PHYSICAL REVIEW E 89, 012143 (2014)
Thermodynamic feature of a Brownian heat engine operating between two heat baths Mesfin Asfaw Department of Physics and Astronomy, California State University Northridge, California 91330-8268, USA (Received 23 September 2013; revised manuscript received 22 November 2013; published 29 January 2014) A generalized theory of nonequilibrium thermodynamics for a Brownian motor operating between two different heat baths is presented. Via a simple paradigmatic model, we not only explore the thermodynamic feature of the engine in the regime of the nonequilibrium steady state but also study the short time behavior of the system for either the isothermal case with load or, in general, the nonisothermal case with or without load. Many elegant thermodynamic theories can be checked via the present model. Furthermore the dependence of the velocity, the efficiency, and the performance of the refrigerator on time t is examined. Our study reveals a current reversal due to time t. In the early system relaxation period, the model works neither as a heat engine nor as a refrigerator and only after a certain period of time does the model start functioning as a heat engine or as a refrigerator. The performance of the engine also improves with time and at steady state the engine manifests a higher efficiency or performance as a refrigerator. Furthermore the effect of energy exchange via the kinetic energy on the performance of the heat engine is explored. DOI: 10.1103/PhysRevE.89.012143
PACS number(s): 05.40.−a
I. INTRODUCTION
Brownian heat engines are tiny microscopic heat engines which exhibit the same thermodynamic principles as macroscopic heat engines. However, the physics of such engines is quite different from that of macroscopic engines; since these motors operate at the submicron scales, thermal fluctuation plays a decisive role in their transport features. In order to get a basic understanding regarding the characteristics and working principles of such thermal engines, several theoretical and experimental works have been carried out [1–11]. Particularly, the performance as well as the transport property of a Brownian heat engine driven by a spatially varying temperature has been intensively studied not only at a quasistatic limit but also with the motor operating within a steady state regime [12–21]. Most of these studies have verified that, at a quasistatic limit, the efficiency and the performance of the refrigerator approach the Carnot efficiency and the performance of the Carnot refrigerator as long as the heat exchange via kinetic energy is neglected. Recently, we considered a Brownian heat engine that modeled particle hopping in a one-dimensional periodic ratchet potential that coupled with a linearly decreasing background temperature. The numerical and exact analytical results revealed that, even if the heat exchange via the kinetic energy is neglected, the efficiency of such a Brownian heat engine approaches the efficiency of an endoreversible heat engine at quasistatic limit [21]. On the other hand, when the heat exchange via the kinetic energy is included, the Carnot efficiency and Carnot refrigerator are unattainable regardless of any parameter choice as there is irreversible heat flow from the hot to the cold reservoirs [16,20,22,23]. So far most of the previous works have addressed how a Brownian heat engine behaves with either a steady state regime or a quasistatic limit and, to best of my knowledge, the role of time on the performance of such motors has never been explored. Indeed, real motors take a finite time to accomplish any task and in order to propose the extents and limits to be considered in the design of actual Brownian heat engines their finite time thermodynamic features have to be addressed. In this work, we address this issue by obtaining exact time1539-3755/2014/89(1)/012143(10)
dependent solutions for the model system considered in the work [13]. This in turn enables us to investigate not only the long time property (steady state) but also the short time behavior of the system. We show that the performance of the engine is quite sensitive to the time t. In the early relaxation period, the model acts as neither a heat engine nor a refrigerator. When t further increases, depending on the parameter choice the model may work as a heat engine or as a refrigerator. The performance of the engine also improves with time and at steady state the engine manifests a higher efficiency or the performance of a refrigerator. Our analysis indicates that when the heat exchange due to the kinetic energy is omitted Carnot refrigerator and Carnot efficiency are achievable only if the system operates quasistatically in a steady state regime. Far from steady state, the efficiency and performance of the engine are far less than the Carnot efficiency or the performance of the Carnot refrigerator, showing that the performance of the engine has a nontrivial dependence on time t. On the other hand, when the energy exchange via kinetic energy is taken into account, we show that the Brownian motor always operates irreversibly. For the model system considered in this work, the system sustains a nonzero velocity as long as a distinct temperature difference between the hot and cold reservoirs is retained or for the isothermal case when a nonzero external force is exerted. Exploring how the velocity depends on time t is vital as it gives us a clue of how to manipulate the mobility of a single monomer and it also serves as a basic paradigm to understand the nonequilibrium transport features of multicomponent systems. In this work, we show that the magnitude and the direction of the velocity are dictated by t, which implies that the mobility of the particle can be manipulated via time t. The model presented in this work also serves as a basic tool for a better understanding of the nonequilibrium statistical physics not only in the regime of the nonequilibrium steady state (NESS) but also at any time t. To date, most of the previous work has dealt with exploring the thermodynamic properties of the nonequilibrium system at a steady state or
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in quasistatic regimes [24–27]. Thus in the present work, we explore the short time behavior of the system for either the isothermal case with load or, in general, the nonisothermal case with or without load. Many elegant mathematical theories can be independently checked via the present model [24,27]. For instance, we show that the entropy balance equation [24] is always satisfied for any parameter choice. Moreover, the first and second laws of thermodynamics are rewritten in terms of the model parameters. Several thermodynamic relations are also uncovered based on the exact analytic results. At this point we emphasize that the results obtained in this work are model independent. The rest of the paper is organized as follows: in Sec. II, we present the model. In Sec. III, we explore the thermodynamic features of the motor. In Sec. IV, we study the dependence of the efficiency, the performance of the refrigerator, and the velocity on the model parameters. We show that the velocity, the efficiency, and the performance of the refrigerator are sensitive to time t. In Sec. V, we explore the influence of heat transfer via the kinetic energy on the performance of the engine. In Sec. VI we present a summary and conclusion. II. THE MODEL
The model system considered in this work is similar to that in our recent work [13]. However, in this work we obtain a general time-dependent solution which allows us to explore the short and long time behavior of the system. Before we calculate the exact time-dependent probability distribution, let us briefly revise the model. Consider a single Brownian particle which undergoes a biased random walk along a one-dimensional discrete ratchet potential with load Ui = E[imod3 − 1] + if d that coupled with the temperature Th , if E[imod3 − 1] = 0; Ti = Tc , otherwise;
(1)
has spacing d, and in one cycle the particle walks a net displacement of three lattice sites. The Brownian particle retains a unidirectional motion as long as a distinct temperature difference between the hot and cold baths is retained or for the isothermal case when a nonzero external force is applied. Its jumping probability from one lattice to the other site is also uniquely detected by the amount of energy it surmounts and the heat reservoir to which it is coupled. Hence the jump probability for the particle to hop from site i to i + 1 is given by e−E/kB Ti , where E = Ui+1 − Ui and is the probability of attempting a jump per unit time. The parameter kB designates the Boltzmann constant, and hereafter kB , , and d are considered to be unity. When the particle undergoes a biased random walk, it is assumed that first it decides which way to jump (backward or forward) with equal probability obeying the metropolis algorism. Accordingly, when E 0, the jump definitely takes place while when E > 0 the jump takes place with the probability exp(−E/Ti ). The dynamics of the model exhibits behavior identical to that of a spin-1 particle system [13]. The master equation which governs the system dynamics is given by dPn (Pnn pn − Pn n pn ) ,n,n = 1,2,3, = (3) dt n=n where Pn n is the transition probability rate at which the system, originally in state n, makes the transition to state n . Here Pn n is given by the Metropolis rule. For instance, P21 = 12 e−(E+f )/Tc , P12 = 12 , P32 = 12 e−(E+f )/Th , and P23 = 12 . The rate equation for the model can then be expressed as the matrix equation d p = Pp, where p = (p1 ,p2 ,p3 )T . P is a 3 by 3 matrix which dt is given by ⎞ ⎛ 2 2 ⎜ P=⎜ ⎝
(2)
as shown in Fig. 1. Here E > 0, f denotes the load and i is an integer that runs from −∞ to ∞. The parameters Th and Tc denote the temperature for the hot and cold reservoirs, respectively. Moreover, the one-dimensional lattice
−μa −μ 2a μa 2 μ2 2a
1 2 −1−νb 2 νb 2
1 2 1 2
⎟ ⎟ ⎠
(4)
−1
as long as 0 < f < 2E/d. Here μ = e−E/Tc , ν = e−E/Th , a = e−f d/Tc , and b = e−f d/Th . Note
that the sum of each column of the matrix P is zero; m Pmn = 0 which
reveals that the total probability is conserved: (d/dt) n pn = d/ = 0. dt(1T · p) = 1T · (Pp) For the particle which is initially situated at site i = 2, the time-dependent normalized probability distributions after some algebra are given by a(2 + νb) μ[μ + (a 2 + μ)νb] (a+a 2 μ+μ2 )t a(−1 + aμ) −1 + , + c2 e− 2a −μ2 + aνb
p1 (t) = c1
p2 (t) = −c3 e FIG. 1. (Color online) Schematic diagram for a Brownian particle walking along a discrete ratchet potential with load. Sites with red circles (situated at i = 2,5, . . .) are coupled to the hot reservoir (Th ) while sites with blue circles (located at i = 1,3,4, . . .) are coupled to the cold reservoir (Tc ). Site 2 is labeled explicitly and d is the lattice spacing. 012143-2
+ c1
1 2 t(−2−νb)
− c2
a e−
(a+a 2 μ+μ2 )t 2a
−μ2
(−1 + aμ) + aνb
(2a 2 + μ) , μ + (a 2 + μ)νb
p3 (t) = c1 + c2 e−
(a+a 2 μ+μ2 )t 2a
(5)
(6) 1
+ c3 e 2 t(−2−νb) ,
(7)
THERMODYNAMIC FEATURE OF A BROWNIAN HEAT . . .
PHYSICAL REVIEW E 89, 012143 (2014)
where μ[μ + (a + μ)νb] , (a + a 2 μ + μ2 )(2 + νb) 2
c1 = c2 = −
(a +
a2μ
+
a
−1 +
μ2 )
a(−1+aμ) −μ2 +aνb
(8) ,
(9)
μ(μ + a 2 νb + μνb) c3 = − (a + a 2 μ + μ2 )(2 + νb) a (10) + .
2 2 (a + a μ + μ ) − 1 + a(−1+aμ) −μ2 +aνb
Note that 3i=1 pi (t) = 1, revealing the probability distribution is normalized. In the limit of t → ∞, we recapture the steady state probability distributions a , a + a 2 μ + μ2
(11)
p2s =
μ(2a 2 + μ) , (a + a 2 μ + μ2 )(2 + bν)
(12)
p3s =
μ(μ + b(a 2 + μ)ν) (a + a 2 μ + μ2 )(2 + bν)
(13)
p1s =
˙ c (t) evolve in time to their ˙ h (t) and Q As t → ∞, both Q corresponding steady state values:
μ baν − μa s ˙ h = (E + f ) Q (18)
2 2(2 + νb) 1 + aμ + μa and ˙ sc Q
3 [Vi+ (t) − Vi− (t)]
(19)
For the isothermal system (uniform temperature T ) which satisfies the detailed balance condition, the relations among the internal energy U (t), the entropy S(t), and the free energy F (t) is well known and can be written as dU [pi (t)] = −T hd (t) = −Hd (t) dt Pj i , (pi Pj i − pj Pij ) ln = −T Pij i>j dF [pi (t)] = −T ep (t) = −EP (t) dt pi Pj i , (pi Pj i − pj Pij ) ln = −T pj Pij i>j
= (p1 P21 − p2 P12 ) + (p2 P32 − p3 P23 )
dS(t) = ep (t) − hd (t), dt
(14)
Exploiting Eq. (14), one can see that the particle retains a unidirectional current when f = 0 and Th > Tc . For the isothermal case Th = Tc , the system sustains a nonzero velocity in the presence of load f = 0 as expected. Moreover, when t → ∞, the velocity V (t) increases with t and approaches steady state velocity:
μ baν − μa s (15) V =3
2 . 2(2 + νb) 1 + aμ + μa Careful analysis also reveals that the heat per unit time taken from the hot reservoir has the form ˙ h (t) = (E + f )(p2 P32 − p3 P23 ) Q P32 . = Th (p2 P32 − p3 P23 ) ln P23
.
(20)
(21)
and the corresponding fundamental entropy balance equation is given by
i=1
+ (p3 P13 − p1 P31 ).
μ2 a
III. THE THERMODYNAMICS FEATURES OF THE MOTOR
shown in Ref. [13]. Now let us analyze the velocity for the particle. The velocity V (t) at any time t is the difference between the forward Vi+ (t) and the backward Vi− (t) velocities at each site i: V (t) =
μ baν − μa = (E − 2f )
2(2 + νb) 1 + aμ +
(16)
On the other hand, the heat per unit time given to the cold reservoir is given by ˙ c (t) = (E + f )(p2 P12 − p1 P21 ) + (2E−f )(p3 P13 − p1 P31 ) Q P12 = Tc (p2 P12 − p1 P21 ) ln P21 P13 + Tc (p3 P13 − p1 P31 ) ln . (17) P31
(22)
where S is the Gibbs entropy given by S[pi (t)] = −
N
pi ln pi .
(23)
i=1
Here hd (t) and ep (t) designate the term which is related to the heat dissipation rate and the instantaneous entropy production (ep > 0), respectively. The entropy balance equation (22) can be rewritten as S T (t) = Ep (t) − Hd (t), where S T (t) = T dS(t)/dt. Recently for the isothermal system which is driven out of equilibrium, the above relations have been extended via a phonological approach [27]. Next, we examine whether the well-known thermodynamic relations, which are valid for an equilibrium system, are still obeyed for the model system which is driven out of equilibrium due to an inhomogeneous thermal arrangement, Th = Tc , or a nonzero external load f = 0. For the three-state model we present, the Gibbs entropy S is given by S[pi (t)] = −
3
pi ln pi .
(24)
i=1
For our nonequilibrium system where Th = Tc and f = 0, regardless of our parameter choice, we find that the fundamental
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entropy balance equation dS(t) = ep (t) − hd (t) (25) dt is always satisfied at any time t. Moreover hd (t) is found to satisfy the relation ˙ h (t) Q ˙ c (t) −Q + . hd (t) = Th Tc
˙ h (t) + Q ˙ c (t) Hd (t) = −Q Pj i . = Tj (pi Pj i − pj Pij ) ln Pij i>j
(27)
Equation (27) is notably different from Eq. (20) due to the term Tj . A similar relation has been used by Ge and Qian for the isothermal case [27]. We rewrite Eq. (27) as Pj i Tj (pi Pj i − pj Pij ) ln Hd (t) = Pij i>j pi Pj i = Tj (pi Pj i − pj Pij ) ln pj Pij i>j pi − Tj (pi Pj i − pj Pij ) ln pj i>j = Ep (t) − S T (t),
Ep (t) =
i>j
and S (t) = T
(28)
Tj (pi Pj i − pj Pij ) ln
i>j
pi Pj i pj Pij
pi Tj (pi Pj i − pj Pij ) ln pj
(29)
.
(30)
Now we have the entropy balance equation S T (t) = Ep (t) − Hd (t) for our model system. Note that for the isothermal case Eqs. (28), (29), and (30) converge to T hd (t), T ep (t), and , respectively. All the above analysis indicates S T (t) = T dS(t) dt that Ep (t), Hd (t), and S T (t) contain a term, (p2 P32 − p3 P23 ), which is associated with the rate of heat taken out of the hot reservoir and two terms, (p2 P12 − p1 P21 ) and (p3 P13 − p1 P31 ), which are associated with the rate of heat given to the cold reservoir. On the other hand, the total internal energy U (t) is the sum of the internal energies [28] U [pi (t)] =
3
dU [Pi (t)] =− (pi Pj i − pj Pij )(ui − uj ) dt i>j ˙ h (t) − Q ˙ c (t) − W (t) =Q = −[Hd (t) + f V (t)].
(26)
In order to relate the free energy dissipation rate with Ep (t) and Hd (t) let us now introduce Hd (t) for the model system we considered. The heat dissipation rate is given by Hd (t) = ˙ h (t) + Q ˙ c (t). Using Eqs. (16) and (17), one finds −Q
where
W (t) = f V (t) against the external load. Hence we verify the first law of thermodynamics:
pi ui = p1 (t)(−E) + p3 (t)(E).
(31)
i=1
As the particle walks along the reaction coordinate, it receives some heat from the hot reservoir and gives part of it to the cold bath. The remaining heat will be spent to do some work
(32)
Note that a similar relation is derived in the work [28] for the isothermal case. Next let us find the expression for the free energy dissipation rate dF /dt where the free energy F satisfies the relation F > 0 and dF /dt < 0. For the isothermal case, the free energy is given by F = U − T S and we can still adapt this relationship to the nonisothermal case to write dF (t) dU = − S T (t). dt dt
(33)
Substituting Eqs. (27) and (29) in Eq. (32) leads to dF (t) dU (t) + Ep (t) = + Hd (t) = −f V (t), dt dt
(34)
which is the second law of thermodynamics. Note that in i (t)] the absence of load, dU [p = −Hd (t) and consequently dt Ep (t) = − dFdt(t) . We want to emphasize that, in the presence of external force, the velocity approaches zero V (t) = 0 in the vicinity of the stall force. The general expression for the stall force is time dependent [see for example Fig. 3(a)]. Since it is lengthy, we do not present it. However, at steady state the stall force reduces to
E TThc − 1 f = Th (35) . 2 Tc + 1 We evaluate Ep near the stall force and find that only in the steady state regime does Ep = 0. This clearly shows that the engine operates reversibly only in the steady state regime. Far from the steady state regime (even in the vicinity of the stall force), Ep > 0, which is expected as the engine operates irreversibly. On the other hand, for the nonisothermal case without load, Ep = 0 in the steady state regime in the limit of E → 0 (quasistatic limit). For the isothermal case without load (the system satisfies the detailed balance condition), Ep = 0 at stationary state (for large t). IV. MORE ON THE ENERGETICS OF THE HEAT ENGINE
In this section we explore the dependence of the velocity, the efficiency, and the performance of the refrigerator on the system parameters. Before exploring the dependence of velocity on t, let us first introduce the dimensionless quantities = E/Tc , λ = f d/Tc and τ = TThc − 1. We also introduce the dimensionless time t¯ = t and hereafter the bar will be dropped. Note that is the probability that the particle will attempt a jump per unit time and in this work it is considered to be unity.
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PHYSICAL REVIEW E 89, 012143 (2014)
A. The particle’s mobility
In this work, we show that the magnitude and the direction of the velocity are dictated by t. Let us now investigate how the velocity of the Brownian particle depends on the system’s parameters. Our analysis indicates that the steady state velocity of the particle is positive when Th = Tc and f = 0. For the isothermal case, V < 0 as long as f > 0. In general for Th = Tc and f > 0, the system exhibits fascinating dynamics where V > 0 when the load is less than the stall force f , f < f , and V < 0 if f > f . This clearly suggests that the mobility of the particle can be manipulated by varying the external force. On the other hand, the short time behavior of the velocity is quite sensitive to time t. In this case the magnitude and the direction of the velocity are dictated by t. This can be notably appreciated by looking at Fig. 2(a). The figure depicts that, for t < 2.04, the net particle flow is from the cold to the hot reservoirs for fixed τ = 5.0, λ = 0.1, and = 1.6. As time increases, the magnitude of V decreases and stalls at t = 2.04. As time further steps up, the particle current gets reversed and the particle moves from the hot to the cold reservoirs until its velocity saturates to a constant value. The plot of V as a function of shows that the particle manifests a peak velocity at a particular barrier height, max . One can note that the engine operates with maximum power at this particular value of max . The potential max at which the velocity of the particle is maximum shifts towards the right when time increases, as can be readily seen in Fig. 2(b). Note
that in the system we consider, the left and the right sides of the sawtooth potential are coupled with the hot and the cold baths, respectively. For such a system, positive velocity exhibits that the net flux of the particle is from the hot to the cold reservoirs. The same Fig. 2(b) depicts that the steady state velocity approaches zero for large , which is expected since for a high potential barrier the Brownian particle encounters difficulty in jumping the ratchet potential. On the contrary, for small t, the particle attains a negative velocity for a large barrier height. B. The efficiency and performance of the refrigerator
As discussed before, so far most of the previous works have explored the operation regimes of the heat engine at steady state. Exploring the short time behavior of the system is vital as it gives us a complete understanding regarding the effect of time on the performance of the engine. Hence in this section, the dependence of the efficiency η and Pref on time t is explored. As mentioned before, in the presence of an external load f = 0, the rate of work done against the load λ is given ˙ h (t) − by W˙ (t) = λV , and only in the limit t → ∞, λv = Q ˙ c (t). Within the regime where the model works as a heat Q engine, the efficiency is given by η(t) =
0.02 0.04
0.05
V
V
0.00
0.10 0.12
0
5
10 t
15
20
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
Λ
(b) 0.04
0.04
0.03
0.02
0.02
0.00
Qh
V
0.06 0.08
0.10
(b)
(36)
(a) 0.02 0.00
(a)
0.15
λV W˙ = . ˙ ˙h Qh Q
0.01 0.00
0.02
0.01
0.04
0.02
0
2
4
Ε
6
8
0.0
10
FIG. 2. (Color online) (a) The velocity V as a function of t for a given = 1.6. (b) V versus . The green (dashed) line is for t = 20 while the red (solid) line is for t = 4. In both panels (a) and (b), τ = 5.0 and λ = 0.1.
0.2
0.4
Λ
FIG. 3. (Color online) (a) The velocity V as a function of λ. (b) Q˙ h (t) versus λ. In both panels (a) and (b), τ = 2.0 and = 2.0. The dotted, dashed, dot-dashed, and solid lines stand for the line plotted by fixing t = 2.0, t = 4.0, t = 6.0, and t = 8.0, respectively.
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(a) 0.5
1.0
0.4
Η
0.8
0.3 0.2
0.4
0.1 0.10
Λ
0.6
0.15
0.20
0.25
0.30
0.35
Λ (b) 0.5
0.2
0.4 5
10
15
20
25
30
0.3
Η
t
0.2
FIG. 4. (Color online) The phase space λ and t for parameter choices = 2.0 and τ = 2. The red shaded region (the lower region) depicts the regime where the model acts as a heat engine. The region where the model works as a refrigerator is marked with blue (the top region) while the white region depicts the regime where the model acts neither as a heat engine nor as a refrigerator.
The hot reservoir serves as the main source of energy and the model acts as a heat engine when heat energy is taken out of the hot reservoir so that the system can sustain a positive unidirectional current. This implies the system may act as a heat engine when Q˙ h (t) > 0 and V > 0. Our analysis indicates ˙ h (t) > 0 and that in the early relaxation time of the particle Q V < 0 or vice versa. In this regime the model acts neither as a heat engine nor as a refrigerator. On the other hand the engine acts as a refrigerator when heat ˙ c (t) < 0 is delivered from the cold to the hot reservoir, i.e., Q and V < 0. In this regime, the performance of the refrigerator is given by Pref (t) =
˙c ˙c Q Q . = ˙ (λV ) W
(37)
Once again, in the early system relation period, depending ˙ c (t) < 0 and V > 0 or vice on the choice of the parameters, Q versa. In this case the model acts neither as a heat engine nor as a refrigerator. In addition, for the Metropolis algorism to hold true, we limit the range of λ, 0 < λ < 2. Thus for λ > 2, the model does not function as a heat engine or as a refrigerator. ˙ h (t) In order to clarify these regimes, the plots of V and Q as a function of λ are depicted in Figs. 3(a) and 3(b) for parameter choices of τ = 2.0 and = 2.0. In the figures the dotted, dashed, dot-dashed, and solid lines are evaluated by fixing t = 2.0, t = 4.0, t = 6.0, and t = 8.0, respectively. The figures exhibit that for t = 2.0 (in the system early relaxation ˙ h (t) > 0, V < 0 and in this regime the model period), even if Q acts neither as a heat engine nor as a refrigerator. Then as time increases (t = 4.0), the system does not act as a heat engine for ˙ h (t) < 0 and V < 0 some values of λ. However for large λ, Q
0.1 0.0 1.0
1.5
2.0
2.5
3.0
Ε FIG. 5. (Color online) (a) The efficiency η as a function of λ for fixed τ = 2.0 and = 2.0. (b) η versus for fixed τ = 2.0 and λ = 0.2. In both panels (a) and (b), the lowest blue line, the intermediate red line, and the upper black line are plotted by fixing t = 8.0, t = 10.0, and t = 30.0, respectively.
and the model functions as a refrigerator. Then as t increases (t = 6.0 and t = 8.0), depending on the magnitude of the load, the model acts as a heat engine or as a refrigerator. One can note that the stall force increases as t increases. In Fig. 4, we plot the phase diagram showing the regime where the model acts as a heat engine or a refrigerator in the parameter space of load and time. Within the regimes where the model functions as a heat engine, the efficiency increases towards its steady state efficiency, η=
3λ , ( + λ)
(38)
and similarly the quasistatic efficiency increases with time towards its Carnot efficiency, ηC =
τ −1 . τ
(39)
Figure 5(a) depicts the plot of η as a function of λ. The figure shows that for t = 30, the system is already at steady state and the efficiency monotonously increases with λ. Near the stall force λ = 0.4, η = 0.5, which is the Carnot efficiency since τ = 2.0. For the cases t = 8.0 and t = 10.0, the stall forces are λ = 0.326 and λ = 0.375, respectively. Unlike the steady
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0.000
0.6
0.002
W
(a) 0.002
Pref
(a) 1.0
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0.2 0.4
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14
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Ε
Λ
(b)
(b) 1.0
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0.8
1.60 1.55
Εopt
0.6
Pref
1.0
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6
1.0
state case, near the stall force η = 0, showing (for small t) the engine operates irreversibly even at stall force. Far from the stall force η exhibits a pronounced peak, ηopt , at a certain optimal load, λopt . ηopt and λopt increase with t. The plot of η versus shows that the steady state efficiency (t = 30) decreases from its Carnot efficiency as increases. For small t, near the stall force η = 0. Far from the stall force η manifests a peak at certain opt . opt shifts to the left as t increases [see Fig. 5(b)]. We also evaluate Ep near the stall force and find that Ep = 0 for the steady state case only. For a large load, the temperature is not strong enough to renormalize the effect of the load and hence the velocity of the particle becomes negative as long as the load is larger than the stall force. In this case the model may act as a refrigerator and the performance of the refrigerator increases with t towards its steady state performance as a refrigerator: Pref =
( − 2λ) . 3λ
(40)
The quasistatic performance of the refrigerator also steps up with time towards its Carnot refrigerator: C Pref =
1 . τ −1
(41)
10
12
t
Ε FIG. 6. (Color online) (a) Pref as a function of λ for fixed τ = 2.0 and = 2.0. The lowest blue line, the intermediate red line, and the upper black line are plotted by fixing t = 15, t = 17, and t = 30, respectively. (b) Pref versus for fixed τ = 2.0 and λ = 0.2. The lowest blue line, the intermediate red line, and the top black line are plotted by fixing t = 8.0, t = 10.0, and t = 30.0, respectively.
8
FIG. 7. (Color online) (a) The power output W˙ as a function of for fixed τ = 2.0 and λ = 0.2. In both figures, the lowest blue line, the intermediate red line, and the upper black line are plotted by fixing t = 8.0, t = 10.0, and t = 30.0, respectively. (b) opt versus t for fixed τ = 2.0 and λ = 0.2.
Pref as a function of λ is plotted in Fig. 6(a). The figure depicts for t = 30 that the system is at steady state and that Pref monotonously decreases with λ. Near the stall force λ = 0.4, Pref = 1.0, which is the Carnot refrigerator since τ = 2.0. Far small t (t = 8.0 and t = 10.0), Pref = 0 at stall force. Pref also exhibits a pronounced peak at a certain optimal load, λopt . λopt decreases with t. The plot of Pref as a function of exhibits that Pref (t = 30) increases towards the Carnot refrigerator as increases. For small t, Pref shows a peak at a certain opt . opt shifts to the right as t increases [see Fig. 6(b)]. The power output W˙ as a function of the barrier height depicts that such a thermal engine delivers finite power while operating irreversibly. The output power increases with t and has a peak at a certain opt . This corresponds to the point in the parameter space where the engine operates with maximum power [see Fig. 7(a)]. opt also explicitly depends on t. As t steps up, opt increases and saturates to a constant value as shown in Fig. 7(b). A closer look at Fig. 7(a) once again reveals that the power output is negative (which implies a positive power input) when < 1.013, < 1.002, and < 1.0 for parameter choices of t = 8, t = 10, and t = 30, respectively. This implies that for small the power output is negative (positive power input) and the heat device works as a refrigerator [see Fig. 8(a)]. In order to clarify this, the phase diagram [Fig. 8(a)] is plotted with the
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the maximum performance of a refrigerator are quite different in optimization in the working regimes. For instance, as shown in Fig. 6(b), the performance of the refrigerator clearly exhibits an optimal value at certain opt as long as t is small. However, in the regime where the model works as a refrigerator [see Fig. 8(b)], the power input does not manifest an optimal value for any . From the above analyses one can deduce that for the engine to operate reversibly, it not only have to be in the steady state regime but also has to operate near the stall force. This implies that for small t Carnot efficiency or Carnot refrigerator is unattainable as there is irreversible heat flow from the hot to the cold reservoirs. So far we consider only the heat flow due to the change of the potential energy. If one includes the heat flow due to the kinetic energy, Carnot efficiency or Carnot refrigeration is unachievable for any parameter choice as is discussed in the next section.
(a) 4.0 3.5 3.0
Ε
2.5 2.0 1.5 1.0 0.5 5
(b)
10
15
t
20
25
30
0.00 0.02
W
0.04 0.06 0.08 0.10 0.12 0
2
4 Ε
6
8
FIG. 8. (Color online) (a) The phase space and t for parameter choices λ = 0.2 and τ = 2. The red shaded region (the upper region) depicts the regime where the model acts as a heat engine. The region where the model works as a refrigerator is marked with blue (the lower region) while the white region depicts the regime where the model acts neither as a heat engine nor as a refrigerator. (b) The power input W˙ as a function of for fixed τ = 2.0 and λ = 0.6. In the figure, the blue, red, and black lines stand for the line plotted by fixing t = 15.0, t = 17.0, and t = 30.0, respectively.
same parameter choice as in Fig. 7(a). As shown in Figs. 7(a) and 8(a), in a concrete range of values, the heat device works as a heat engine, i.e., when > 1.013, > 1.002, and > 1.0 for parameter choices of t = 8, t = 10, and t = 30, respectively. On the other hand, for small t (t = 8 and t = 10) and large values, the power output is once again negative. In this regime as shown in the phase diagram (Fig. 8), the model works neither as a heat engine nor as a refrigerator. At this point we want to emphasize that the efficiency and power output exhibit optimal values at certain opt [see Figs. 5(a) and 7(a)]. However the magnitude of opt (at which the efficiency is maximum) is quite different from that of the power output. The above analysis shows that, in general, the maximum power and the maximum efficiency as well as
V. IRREVERSIBILITY DUE TO THE HEAT FLOW VIA THE KINETIC ENERGY
In this section, we investigate the effect of heat exchange via the kinetic energy on the performance of the engine. This effect has been discussed already in Refs. [16,20,22,23]. However, most of these works address how the performance of the engine behaves in a steady state regime. Instead of studying only the long time behavior of the system, we would like to explore the model system for any time t. Our analytic result reveals that the efficiency or the performance of the refrigerator never goes to the Carnot efficiency or Carnot refrigerator at the quasistatic limit. When the particle moves from the hot to the cold heat baths, in one cycle, a 12 (τ − 1) amount of energy is transferred from the hot to the cold heat baths via kinetic energy [16,20,22,23]. Similarly, when the engine acts as a refrigerator, the net flow of the particle is from the cold to the hot reservoirs. In this case, due to particle recrossing between the hot and the cold reservoirs, heat is leaking from the hot to the cold reservoir at a magnitude of 12 (τ − 1). The rate of the heat flow (from the hot to the cold heat baths due to kinetic energy) is given ˙ ∗ (t) = 1 kB (Th − Tc )J . Hence the total heat per unit time by Q h 2 taken from the hot reservoir has the form 1 ˙ irr ˙ Q h (t) = Qh (t) + 2 (τ − 1)J,
(42)
while the total heat per unit time from the cold reservoir is given by 1 ˙ irr ˙ Q c (t) = Qc (t) + 2 (τ − 1)J.
(43)
Note that, when the engine functions as a refrigerator, Q˙ irr c (t) = ˙ c (t) − 1 (1 − τ )J . Here J = V /3. Q 2 ˙ irr (t) < η and P ∗ = Q ˙ irr The efficiency η∗ = λV /Q c (t)/ h ref λV < Pref . Before we explore the general dependence of η∗ ∗ and Pref on system parameters, let us determine the efficiency and performance of the refrigerator at steady state. At steady state
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η∗ =
6λ 2 + 2λ − 1 + τ
(44)
THERMODYNAMIC FEATURE OF A BROWNIAN HEAT . . .
PHYSICAL REVIEW E 89, 012143 (2014)
and
(a) 0.40 2 − 4λ + 1 − τ . 6λ
τ −1 In the quasistatic limit (λ → 1+2τ ), the steady state efficiency takes the form
η∗ =
τ −1 = ηC , τ
0.35
(45)
0.30
Η irr
∗ = Pref
0.20
(46)
0.15
where 6 = 2(3 + τ ) −
1 τ
−1
.
2(3 − τ 2 ) + τ + 1 . 6
(b)
0.30
0.35
0.5 0.4 0.3
(48)
where
0.25
Λ
P ref
1 ∗ C Pref
= Pref (t) =
, τ −1
0.10 0.20
(47)
Here 0 < < 1, revealing that the efficiency never approaches the Carnot efficiency ηC . Hence η∗ < η. In the ∗ converges to quasistatic limit, the steady state Pref
=
0.25
0.2 0.1
(49)
0.0
C Pref
Note that 0 < < 1, revealing that Carnot refrigeration is unattainable. We need to justify the above analysis via different approaches and retrieve the results shown above without considering the heat dissipation rate. When the engine functions as a heat engine, in one cycle, the particles takes a Q∗h = ( + λ) + 12 (τ − 1) amount of heat from the hot bath. The heat flow to the cold reservoir is given by Q∗c = ( − 2λ) + 1 (τ − 1). On the other hand when the heat engine acts as a 2 refrigerator the net heat flow out of the cold bath is given ∗ by Q∗c = ( − 2λ) − 12 (τ − 1) [20]. Evaluating η∗ and Pref , we find them to be the same as Eqs. (44) and (45). In the quasistatic limit, we retrieve Eqs. (46) and (48). Far from steady state, we evaluate the behavior of η∗ and ∗ Pref . Figure 9(a) is plotted by taking the same parameter choice as that in Fig. 5(a). However, η∗ is much less than the corresponding η, showing irreversible heat from the hot ∗ to the cold reservoir. Pref as a function of load is plotted in Fig. 9(b) by taking the same parameter choice as in Fig. 6(a). ∗ The figure depicts that Pref < Pref .
VI. SUMMARY AND CONCLUSION
In this work, we present a simple model which has few ingredients. The exact analytical results reveal the sensitivity of the performance of the thermal engine to the time t. Its operation regimes are dictated by the operation time t. In the early particle relaxation period (small t), the engine neither acts as a heat engine nor as a refrigerator. This is because, when the system relaxation time is less than the time that the engine needs to perform work, the energy taken from the hot bath dissipates without doing any work. When t further increases, depending on the parameter choice, the motor may work as a heat engine or as a refrigerator. Its performance
0.5
0.6 Λ
0.7
0.8
FIG. 9. (Color online) (a) The efficiency η as a function of λ for the same parameter choice as in Fig. 5(a). (b) Pref versus λ for the same parameter choice as in Fig. 6(a).
is also an increasing function of t. Furthermore the engine depicts a higher efficiency or performance as a refrigerator at steady state. Moreover, we show that, when one omits the heat exchange via the kinetic energy, performance as a Carnot refrigerator and Carnot efficiency are attainable only when the system operates quasistatically in the steady state regime. The magnitude and the direction of the velocity are also controlled by t. We stress that the size ratchet potential can be varied on a mesoscopic or macroscopic scale, implying that a reasonable temperature difference can be imposed along the ratchet potential [29]. The dynamics of such a system can be realized experimentally (at least for the continuum case). One makes a negatively charged monomer, then puts the particle within a positively and negatively charged fluidic channel [30]. Part of the fluidic channel is also heated. Since the Brownian particle is negatively charged, it encounters difficulty crossing through the negatively charged part of the channel. However, assisted by the nonuniform thermal background kicks, it ultimately attains a unidirectional particle current. The presence of external force may further influence the direction and the magnitude of the velocity of the particle. In conclusion, in this work we present a simple model which serves as a basic tool for better understanding of nonequilibrium statistical physics not only in the regime of NESS but also at any time t. The present model also serves as
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a tool to check many elegant thermodynamic theories. Based on this exactly solvable model, we have uncovered several thermodynamic relations.
[1] N. Li, F. Zhan, P. H¨anggi, and B. Li, Phys. Rev. E 80, 011125 (2009). [2] N. Li, P. H¨anggi, and B. Li, Europhys. Lett. 84, 40009 (2008). [3] F. Zhan, N. Li, S. Kohler, and P. H¨anggi, Phys. Rev. E 80, 061115 (2009). [4] M. B¨uttiker, Z. Phys. B 68, 161 (1987). [5] N. G. van Kampen, IBM J. Res. Dev. 32, 107 (1988). [6] R. Landauer, J. Stat. Phys. 53, 233 (1988). [7] R. Landauer, Phys. Rev. A 12, 636 (1975). [8] R. Landauer, Helv. Phys. Acta 56, 847 (1983). [9] P. H¨anggi, F. Marchesoni, and F. Nori, Ann. Phys. (Leipzig) 14, 51 (2005). [10] P. H¨anggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009). [11] P. Reimann, R. Bartussek, R. Haussler, and P. H¨anggi, Phys. Lett. A 215, 26 (1996). [12] M. Asfaw and M. Bekele, Eur. Phys. J. B 38, 457 (2004). [13] M. Asfaw and M. Bekele, Phys. Rev. E 72, 056109 (2005). [14] M. Asfaw and M. Bekele, Phys. A 384, 346 (2007). [15] M. Matsuo and S. I. Sasa, Phys. A 276, 188 (2000). [16] I. Der`enyi and R. D. Astumian, Phys. Rev. E 59, R6219 (1999).
ACKNOWLEDGMENT
I would like to thank Y. Shiferaw and M. Bekele for the interesting discussions.
[17] I. Der`enyi, M. Bier, and R. D. Astumian, Phys. Rev. Lett. 83, 903 (1999). [18] J. M. Sancho, M. S. Miguel, and D. D¨urr, J. Stat. Phys. 28, 291 (1982). [19] B. Q. Ai, H. Z. Xie, D. H. Wen, X. M. Liu, and L. G. Liu, Eur. Phys. J. B 48, 101 (2005). [20] M. Asfaw, Eur. Phys. J. B 65, 109 (2008). [21] M. Asfaw, Eur. Phys. J. B 86, 189 (2013). [22] T. Hondou and K. Sekimoto, Phys. Rev. E 62, 6021 (2000). [23] Y. Zhang and J. C. Chen, Eur. Phys. J. B 53, 481 (2006). [24] J. Schnakenberg, Rev. Mod. Phys. 48, 571 (1976). [25] J. Y. Oono and M. Paniconi, Prog. Theor. Phys. 130, 29 (1998). [26] T. Hatano and S. I. Sasa, Phys. Rev. Lett. 86, 3463 (2001). [27] H. Ge and H. Qian, Phys. Rev. E 81, 051133 (2010). [28] J. Parrondo, B. Jimenez de Cisneros, and R. Brito, in Stochastic Processes in Physics, Chemistry and Biology, Lecture Notes in Physics, Vol. 557 (Springer-Verlag, Berlin, 2000), p. 38. [29] F. J¨ulicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997). [30] M. Asfaw, Phys. Rev. E 82, 021111 (2010).
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Thermodynamic feature of a Brownian heat engine operating between two heat baths Mesfin Asfaw Department of Physics and Astronomy, California State University Northridge, California 91330-8268, USA (Received 23 September 2013; revised manuscript received 22 November 2013; published 29 January 2014) A generalized theory of nonequilibrium thermodynamics for a Brownian motor operating between two different heat baths is presented. Via a simple paradigmatic model, we not only explore the thermodynamic feature of the engine in the regime of the nonequilibrium steady state but also study the short time behavior of the system for either the isothermal case with load or, in general, the nonisothermal case with or without load. Many elegant thermodynamic theories can be checked via the present model. Furthermore the dependence of the velocity, the efficiency, and the performance of the refrigerator on time t is examined. Our study reveals a current reversal due to time t. In the early system relaxation period, the model works neither as a heat engine nor as a refrigerator and only after a certain period of time does the model start functioning as a heat engine or as a refrigerator. The performance of the engine also improves with time and at steady state the engine manifests a higher efficiency or performance as a refrigerator. Furthermore the effect of energy exchange via the kinetic energy on the performance of the heat engine is explored. DOI: 10.1103/PhysRevE.89.012143
PACS number(s): 05.40.−a
I. INTRODUCTION
Brownian heat engines are tiny microscopic heat engines which exhibit the same thermodynamic principles as macroscopic heat engines. However, the physics of such engines is quite different from that of macroscopic engines; since these motors operate at the submicron scales, thermal fluctuation plays a decisive role in their transport features. In order to get a basic understanding regarding the characteristics and working principles of such thermal engines, several theoretical and experimental works have been carried out [1–11]. Particularly, the performance as well as the transport property of a Brownian heat engine driven by a spatially varying temperature has been intensively studied not only at a quasistatic limit but also with the motor operating within a steady state regime [12–21]. Most of these studies have verified that, at a quasistatic limit, the efficiency and the performance of the refrigerator approach the Carnot efficiency and the performance of the Carnot refrigerator as long as the heat exchange via kinetic energy is neglected. Recently, we considered a Brownian heat engine that modeled particle hopping in a one-dimensional periodic ratchet potential that coupled with a linearly decreasing background temperature. The numerical and exact analytical results revealed that, even if the heat exchange via the kinetic energy is neglected, the efficiency of such a Brownian heat engine approaches the efficiency of an endoreversible heat engine at quasistatic limit [21]. On the other hand, when the heat exchange via the kinetic energy is included, the Carnot efficiency and Carnot refrigerator are unattainable regardless of any parameter choice as there is irreversible heat flow from the hot to the cold reservoirs [16,20,22,23]. So far most of the previous works have addressed how a Brownian heat engine behaves with either a steady state regime or a quasistatic limit and, to best of my knowledge, the role of time on the performance of such motors has never been explored. Indeed, real motors take a finite time to accomplish any task and in order to propose the extents and limits to be considered in the design of actual Brownian heat engines their finite time thermodynamic features have to be addressed. In this work, we address this issue by obtaining exact time1539-3755/2014/89(1)/012143(10)
dependent solutions for the model system considered in the work [13]. This in turn enables us to investigate not only the long time property (steady state) but also the short time behavior of the system. We show that the performance of the engine is quite sensitive to the time t. In the early relaxation period, the model acts as neither a heat engine nor a refrigerator. When t further increases, depending on the parameter choice the model may work as a heat engine or as a refrigerator. The performance of the engine also improves with time and at steady state the engine manifests a higher efficiency or the performance of a refrigerator. Our analysis indicates that when the heat exchange due to the kinetic energy is omitted Carnot refrigerator and Carnot efficiency are achievable only if the system operates quasistatically in a steady state regime. Far from steady state, the efficiency and performance of the engine are far less than the Carnot efficiency or the performance of the Carnot refrigerator, showing that the performance of the engine has a nontrivial dependence on time t. On the other hand, when the energy exchange via kinetic energy is taken into account, we show that the Brownian motor always operates irreversibly. For the model system considered in this work, the system sustains a nonzero velocity as long as a distinct temperature difference between the hot and cold reservoirs is retained or for the isothermal case when a nonzero external force is exerted. Exploring how the velocity depends on time t is vital as it gives us a clue of how to manipulate the mobility of a single monomer and it also serves as a basic paradigm to understand the nonequilibrium transport features of multicomponent systems. In this work, we show that the magnitude and the direction of the velocity are dictated by t, which implies that the mobility of the particle can be manipulated via time t. The model presented in this work also serves as a basic tool for a better understanding of the nonequilibrium statistical physics not only in the regime of the nonequilibrium steady state (NESS) but also at any time t. To date, most of the previous work has dealt with exploring the thermodynamic properties of the nonequilibrium system at a steady state or
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in quasistatic regimes [24–27]. Thus in the present work, we explore the short time behavior of the system for either the isothermal case with load or, in general, the nonisothermal case with or without load. Many elegant mathematical theories can be independently checked via the present model [24,27]. For instance, we show that the entropy balance equation [24] is always satisfied for any parameter choice. Moreover, the first and second laws of thermodynamics are rewritten in terms of the model parameters. Several thermodynamic relations are also uncovered based on the exact analytic results. At this point we emphasize that the results obtained in this work are model independent. The rest of the paper is organized as follows: in Sec. II, we present the model. In Sec. III, we explore the thermodynamic features of the motor. In Sec. IV, we study the dependence of the efficiency, the performance of the refrigerator, and the velocity on the model parameters. We show that the velocity, the efficiency, and the performance of the refrigerator are sensitive to time t. In Sec. V, we explore the influence of heat transfer via the kinetic energy on the performance of the engine. In Sec. VI we present a summary and conclusion. II. THE MODEL
The model system considered in this work is similar to that in our recent work [13]. However, in this work we obtain a general time-dependent solution which allows us to explore the short and long time behavior of the system. Before we calculate the exact time-dependent probability distribution, let us briefly revise the model. Consider a single Brownian particle which undergoes a biased random walk along a one-dimensional discrete ratchet potential with load Ui = E[imod3 − 1] + if d that coupled with the temperature Th , if E[imod3 − 1] = 0; Ti = Tc , otherwise;
(1)
has spacing d, and in one cycle the particle walks a net displacement of three lattice sites. The Brownian particle retains a unidirectional motion as long as a distinct temperature difference between the hot and cold baths is retained or for the isothermal case when a nonzero external force is applied. Its jumping probability from one lattice to the other site is also uniquely detected by the amount of energy it surmounts and the heat reservoir to which it is coupled. Hence the jump probability for the particle to hop from site i to i + 1 is given by e−E/kB Ti , where E = Ui+1 − Ui and is the probability of attempting a jump per unit time. The parameter kB designates the Boltzmann constant, and hereafter kB , , and d are considered to be unity. When the particle undergoes a biased random walk, it is assumed that first it decides which way to jump (backward or forward) with equal probability obeying the metropolis algorism. Accordingly, when E 0, the jump definitely takes place while when E > 0 the jump takes place with the probability exp(−E/Ti ). The dynamics of the model exhibits behavior identical to that of a spin-1 particle system [13]. The master equation which governs the system dynamics is given by dPn (Pnn pn − Pn n pn ) ,n,n = 1,2,3, = (3) dt n=n where Pn n is the transition probability rate at which the system, originally in state n, makes the transition to state n . Here Pn n is given by the Metropolis rule. For instance, P21 = 12 e−(E+f )/Tc , P12 = 12 , P32 = 12 e−(E+f )/Th , and P23 = 12 . The rate equation for the model can then be expressed as the matrix equation d p = Pp, where p = (p1 ,p2 ,p3 )T . P is a 3 by 3 matrix which dt is given by ⎞ ⎛ 2 2 ⎜ P=⎜ ⎝
(2)
as shown in Fig. 1. Here E > 0, f denotes the load and i is an integer that runs from −∞ to ∞. The parameters Th and Tc denote the temperature for the hot and cold reservoirs, respectively. Moreover, the one-dimensional lattice
−μa −μ 2a μa 2 μ2 2a
1 2 −1−νb 2 νb 2
1 2 1 2
⎟ ⎟ ⎠
(4)
−1
as long as 0 < f < 2E/d. Here μ = e−E/Tc , ν = e−E/Th , a = e−f d/Tc , and b = e−f d/Th . Note
that the sum of each column of the matrix P is zero; m Pmn = 0 which
reveals that the total probability is conserved: (d/dt) n pn = d/ = 0. dt(1T · p) = 1T · (Pp) For the particle which is initially situated at site i = 2, the time-dependent normalized probability distributions after some algebra are given by a(2 + νb) μ[μ + (a 2 + μ)νb] (a+a 2 μ+μ2 )t a(−1 + aμ) −1 + , + c2 e− 2a −μ2 + aνb
p1 (t) = c1
p2 (t) = −c3 e FIG. 1. (Color online) Schematic diagram for a Brownian particle walking along a discrete ratchet potential with load. Sites with red circles (situated at i = 2,5, . . .) are coupled to the hot reservoir (Th ) while sites with blue circles (located at i = 1,3,4, . . .) are coupled to the cold reservoir (Tc ). Site 2 is labeled explicitly and d is the lattice spacing. 012143-2
+ c1
1 2 t(−2−νb)
− c2
a e−
(a+a 2 μ+μ2 )t 2a
−μ2
(−1 + aμ) + aνb
(2a 2 + μ) , μ + (a 2 + μ)νb
p3 (t) = c1 + c2 e−
(a+a 2 μ+μ2 )t 2a
(5)
(6) 1
+ c3 e 2 t(−2−νb) ,
(7)
THERMODYNAMIC FEATURE OF A BROWNIAN HEAT . . .
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where μ[μ + (a + μ)νb] , (a + a 2 μ + μ2 )(2 + νb) 2
c1 = c2 = −
(a +
a2μ
+
a
−1 +
μ2 )
a(−1+aμ) −μ2 +aνb
(8) ,
(9)
μ(μ + a 2 νb + μνb) c3 = − (a + a 2 μ + μ2 )(2 + νb) a (10) + .
2 2 (a + a μ + μ ) − 1 + a(−1+aμ) −μ2 +aνb
Note that 3i=1 pi (t) = 1, revealing the probability distribution is normalized. In the limit of t → ∞, we recapture the steady state probability distributions a , a + a 2 μ + μ2
(11)
p2s =
μ(2a 2 + μ) , (a + a 2 μ + μ2 )(2 + bν)
(12)
p3s =
μ(μ + b(a 2 + μ)ν) (a + a 2 μ + μ2 )(2 + bν)
(13)
p1s =
˙ c (t) evolve in time to their ˙ h (t) and Q As t → ∞, both Q corresponding steady state values:
μ baν − μa s ˙ h = (E + f ) Q (18)
2 2(2 + νb) 1 + aμ + μa and ˙ sc Q
3 [Vi+ (t) − Vi− (t)]
(19)
For the isothermal system (uniform temperature T ) which satisfies the detailed balance condition, the relations among the internal energy U (t), the entropy S(t), and the free energy F (t) is well known and can be written as dU [pi (t)] = −T hd (t) = −Hd (t) dt Pj i , (pi Pj i − pj Pij ) ln = −T Pij i>j dF [pi (t)] = −T ep (t) = −EP (t) dt pi Pj i , (pi Pj i − pj Pij ) ln = −T pj Pij i>j
= (p1 P21 − p2 P12 ) + (p2 P32 − p3 P23 )
dS(t) = ep (t) − hd (t), dt
(14)
Exploiting Eq. (14), one can see that the particle retains a unidirectional current when f = 0 and Th > Tc . For the isothermal case Th = Tc , the system sustains a nonzero velocity in the presence of load f = 0 as expected. Moreover, when t → ∞, the velocity V (t) increases with t and approaches steady state velocity:
μ baν − μa s (15) V =3
2 . 2(2 + νb) 1 + aμ + μa Careful analysis also reveals that the heat per unit time taken from the hot reservoir has the form ˙ h (t) = (E + f )(p2 P32 − p3 P23 ) Q P32 . = Th (p2 P32 − p3 P23 ) ln P23
.
(20)
(21)
and the corresponding fundamental entropy balance equation is given by
i=1
+ (p3 P13 − p1 P31 ).
μ2 a
III. THE THERMODYNAMICS FEATURES OF THE MOTOR
shown in Ref. [13]. Now let us analyze the velocity for the particle. The velocity V (t) at any time t is the difference between the forward Vi+ (t) and the backward Vi− (t) velocities at each site i: V (t) =
μ baν − μa = (E − 2f )
2(2 + νb) 1 + aμ +
(16)
On the other hand, the heat per unit time given to the cold reservoir is given by ˙ c (t) = (E + f )(p2 P12 − p1 P21 ) + (2E−f )(p3 P13 − p1 P31 ) Q P12 = Tc (p2 P12 − p1 P21 ) ln P21 P13 + Tc (p3 P13 − p1 P31 ) ln . (17) P31
(22)
where S is the Gibbs entropy given by S[pi (t)] = −
N
pi ln pi .
(23)
i=1
Here hd (t) and ep (t) designate the term which is related to the heat dissipation rate and the instantaneous entropy production (ep > 0), respectively. The entropy balance equation (22) can be rewritten as S T (t) = Ep (t) − Hd (t), where S T (t) = T dS(t)/dt. Recently for the isothermal system which is driven out of equilibrium, the above relations have been extended via a phonological approach [27]. Next, we examine whether the well-known thermodynamic relations, which are valid for an equilibrium system, are still obeyed for the model system which is driven out of equilibrium due to an inhomogeneous thermal arrangement, Th = Tc , or a nonzero external load f = 0. For the three-state model we present, the Gibbs entropy S is given by S[pi (t)] = −
3
pi ln pi .
(24)
i=1
For our nonequilibrium system where Th = Tc and f = 0, regardless of our parameter choice, we find that the fundamental
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entropy balance equation dS(t) = ep (t) − hd (t) (25) dt is always satisfied at any time t. Moreover hd (t) is found to satisfy the relation ˙ h (t) Q ˙ c (t) −Q + . hd (t) = Th Tc
˙ h (t) + Q ˙ c (t) Hd (t) = −Q Pj i . = Tj (pi Pj i − pj Pij ) ln Pij i>j
(27)
Equation (27) is notably different from Eq. (20) due to the term Tj . A similar relation has been used by Ge and Qian for the isothermal case [27]. We rewrite Eq. (27) as Pj i Tj (pi Pj i − pj Pij ) ln Hd (t) = Pij i>j pi Pj i = Tj (pi Pj i − pj Pij ) ln pj Pij i>j pi − Tj (pi Pj i − pj Pij ) ln pj i>j = Ep (t) − S T (t),
Ep (t) =
i>j
and S (t) = T
(28)
Tj (pi Pj i − pj Pij ) ln
i>j
pi Pj i pj Pij
pi Tj (pi Pj i − pj Pij ) ln pj
(29)
.
(30)
Now we have the entropy balance equation S T (t) = Ep (t) − Hd (t) for our model system. Note that for the isothermal case Eqs. (28), (29), and (30) converge to T hd (t), T ep (t), and , respectively. All the above analysis indicates S T (t) = T dS(t) dt that Ep (t), Hd (t), and S T (t) contain a term, (p2 P32 − p3 P23 ), which is associated with the rate of heat taken out of the hot reservoir and two terms, (p2 P12 − p1 P21 ) and (p3 P13 − p1 P31 ), which are associated with the rate of heat given to the cold reservoir. On the other hand, the total internal energy U (t) is the sum of the internal energies [28] U [pi (t)] =
3
dU [Pi (t)] =− (pi Pj i − pj Pij )(ui − uj ) dt i>j ˙ h (t) − Q ˙ c (t) − W (t) =Q = −[Hd (t) + f V (t)].
(26)
In order to relate the free energy dissipation rate with Ep (t) and Hd (t) let us now introduce Hd (t) for the model system we considered. The heat dissipation rate is given by Hd (t) = ˙ h (t) + Q ˙ c (t). Using Eqs. (16) and (17), one finds −Q
where
W (t) = f V (t) against the external load. Hence we verify the first law of thermodynamics:
pi ui = p1 (t)(−E) + p3 (t)(E).
(31)
i=1
As the particle walks along the reaction coordinate, it receives some heat from the hot reservoir and gives part of it to the cold bath. The remaining heat will be spent to do some work
(32)
Note that a similar relation is derived in the work [28] for the isothermal case. Next let us find the expression for the free energy dissipation rate dF /dt where the free energy F satisfies the relation F > 0 and dF /dt < 0. For the isothermal case, the free energy is given by F = U − T S and we can still adapt this relationship to the nonisothermal case to write dF (t) dU = − S T (t). dt dt
(33)
Substituting Eqs. (27) and (29) in Eq. (32) leads to dF (t) dU (t) + Ep (t) = + Hd (t) = −f V (t), dt dt
(34)
which is the second law of thermodynamics. Note that in i (t)] the absence of load, dU [p = −Hd (t) and consequently dt Ep (t) = − dFdt(t) . We want to emphasize that, in the presence of external force, the velocity approaches zero V (t) = 0 in the vicinity of the stall force. The general expression for the stall force is time dependent [see for example Fig. 3(a)]. Since it is lengthy, we do not present it. However, at steady state the stall force reduces to
E TThc − 1 f = Th (35) . 2 Tc + 1 We evaluate Ep near the stall force and find that only in the steady state regime does Ep = 0. This clearly shows that the engine operates reversibly only in the steady state regime. Far from the steady state regime (even in the vicinity of the stall force), Ep > 0, which is expected as the engine operates irreversibly. On the other hand, for the nonisothermal case without load, Ep = 0 in the steady state regime in the limit of E → 0 (quasistatic limit). For the isothermal case without load (the system satisfies the detailed balance condition), Ep = 0 at stationary state (for large t). IV. MORE ON THE ENERGETICS OF THE HEAT ENGINE
In this section we explore the dependence of the velocity, the efficiency, and the performance of the refrigerator on the system parameters. Before exploring the dependence of velocity on t, let us first introduce the dimensionless quantities = E/Tc , λ = f d/Tc and τ = TThc − 1. We also introduce the dimensionless time t¯ = t and hereafter the bar will be dropped. Note that is the probability that the particle will attempt a jump per unit time and in this work it is considered to be unity.
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A. The particle’s mobility
In this work, we show that the magnitude and the direction of the velocity are dictated by t. Let us now investigate how the velocity of the Brownian particle depends on the system’s parameters. Our analysis indicates that the steady state velocity of the particle is positive when Th = Tc and f = 0. For the isothermal case, V < 0 as long as f > 0. In general for Th = Tc and f > 0, the system exhibits fascinating dynamics where V > 0 when the load is less than the stall force f , f < f , and V < 0 if f > f . This clearly suggests that the mobility of the particle can be manipulated by varying the external force. On the other hand, the short time behavior of the velocity is quite sensitive to time t. In this case the magnitude and the direction of the velocity are dictated by t. This can be notably appreciated by looking at Fig. 2(a). The figure depicts that, for t < 2.04, the net particle flow is from the cold to the hot reservoirs for fixed τ = 5.0, λ = 0.1, and = 1.6. As time increases, the magnitude of V decreases and stalls at t = 2.04. As time further steps up, the particle current gets reversed and the particle moves from the hot to the cold reservoirs until its velocity saturates to a constant value. The plot of V as a function of shows that the particle manifests a peak velocity at a particular barrier height, max . One can note that the engine operates with maximum power at this particular value of max . The potential max at which the velocity of the particle is maximum shifts towards the right when time increases, as can be readily seen in Fig. 2(b). Note
that in the system we consider, the left and the right sides of the sawtooth potential are coupled with the hot and the cold baths, respectively. For such a system, positive velocity exhibits that the net flux of the particle is from the hot to the cold reservoirs. The same Fig. 2(b) depicts that the steady state velocity approaches zero for large , which is expected since for a high potential barrier the Brownian particle encounters difficulty in jumping the ratchet potential. On the contrary, for small t, the particle attains a negative velocity for a large barrier height. B. The efficiency and performance of the refrigerator
As discussed before, so far most of the previous works have explored the operation regimes of the heat engine at steady state. Exploring the short time behavior of the system is vital as it gives us a complete understanding regarding the effect of time on the performance of the engine. Hence in this section, the dependence of the efficiency η and Pref on time t is explored. As mentioned before, in the presence of an external load f = 0, the rate of work done against the load λ is given ˙ h (t) − by W˙ (t) = λV , and only in the limit t → ∞, λv = Q ˙ c (t). Within the regime where the model works as a heat Q engine, the efficiency is given by η(t) =
0.02 0.04
0.05
V
V
0.00
0.10 0.12
0
5
10 t
15
20
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
Λ
(b) 0.04
0.04
0.03
0.02
0.02
0.00
Qh
V
0.06 0.08
0.10
(b)
(36)
(a) 0.02 0.00
(a)
0.15
λV W˙ = . ˙ ˙h Qh Q
0.01 0.00
0.02
0.01
0.04
0.02
0
2
4
Ε
6
8
0.0
10
FIG. 2. (Color online) (a) The velocity V as a function of t for a given = 1.6. (b) V versus . The green (dashed) line is for t = 20 while the red (solid) line is for t = 4. In both panels (a) and (b), τ = 5.0 and λ = 0.1.
0.2
0.4
Λ
FIG. 3. (Color online) (a) The velocity V as a function of λ. (b) Q˙ h (t) versus λ. In both panels (a) and (b), τ = 2.0 and = 2.0. The dotted, dashed, dot-dashed, and solid lines stand for the line plotted by fixing t = 2.0, t = 4.0, t = 6.0, and t = 8.0, respectively.
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(a) 0.5
1.0
0.4
Η
0.8
0.3 0.2
0.4
0.1 0.10
Λ
0.6
0.15
0.20
0.25
0.30
0.35
Λ (b) 0.5
0.2
0.4 5
10
15
20
25
30
0.3
Η
t
0.2
FIG. 4. (Color online) The phase space λ and t for parameter choices = 2.0 and τ = 2. The red shaded region (the lower region) depicts the regime where the model acts as a heat engine. The region where the model works as a refrigerator is marked with blue (the top region) while the white region depicts the regime where the model acts neither as a heat engine nor as a refrigerator.
The hot reservoir serves as the main source of energy and the model acts as a heat engine when heat energy is taken out of the hot reservoir so that the system can sustain a positive unidirectional current. This implies the system may act as a heat engine when Q˙ h (t) > 0 and V > 0. Our analysis indicates ˙ h (t) > 0 and that in the early relaxation time of the particle Q V < 0 or vice versa. In this regime the model acts neither as a heat engine nor as a refrigerator. On the other hand the engine acts as a refrigerator when heat ˙ c (t) < 0 is delivered from the cold to the hot reservoir, i.e., Q and V < 0. In this regime, the performance of the refrigerator is given by Pref (t) =
˙c ˙c Q Q . = ˙ (λV ) W
(37)
Once again, in the early system relation period, depending ˙ c (t) < 0 and V > 0 or vice on the choice of the parameters, Q versa. In this case the model acts neither as a heat engine nor as a refrigerator. In addition, for the Metropolis algorism to hold true, we limit the range of λ, 0 < λ < 2. Thus for λ > 2, the model does not function as a heat engine or as a refrigerator. ˙ h (t) In order to clarify these regimes, the plots of V and Q as a function of λ are depicted in Figs. 3(a) and 3(b) for parameter choices of τ = 2.0 and = 2.0. In the figures the dotted, dashed, dot-dashed, and solid lines are evaluated by fixing t = 2.0, t = 4.0, t = 6.0, and t = 8.0, respectively. The figures exhibit that for t = 2.0 (in the system early relaxation ˙ h (t) > 0, V < 0 and in this regime the model period), even if Q acts neither as a heat engine nor as a refrigerator. Then as time increases (t = 4.0), the system does not act as a heat engine for ˙ h (t) < 0 and V < 0 some values of λ. However for large λ, Q
0.1 0.0 1.0
1.5
2.0
2.5
3.0
Ε FIG. 5. (Color online) (a) The efficiency η as a function of λ for fixed τ = 2.0 and = 2.0. (b) η versus for fixed τ = 2.0 and λ = 0.2. In both panels (a) and (b), the lowest blue line, the intermediate red line, and the upper black line are plotted by fixing t = 8.0, t = 10.0, and t = 30.0, respectively.
and the model functions as a refrigerator. Then as t increases (t = 6.0 and t = 8.0), depending on the magnitude of the load, the model acts as a heat engine or as a refrigerator. One can note that the stall force increases as t increases. In Fig. 4, we plot the phase diagram showing the regime where the model acts as a heat engine or a refrigerator in the parameter space of load and time. Within the regimes where the model functions as a heat engine, the efficiency increases towards its steady state efficiency, η=
3λ , ( + λ)
(38)
and similarly the quasistatic efficiency increases with time towards its Carnot efficiency, ηC =
τ −1 . τ
(39)
Figure 5(a) depicts the plot of η as a function of λ. The figure shows that for t = 30, the system is already at steady state and the efficiency monotonously increases with λ. Near the stall force λ = 0.4, η = 0.5, which is the Carnot efficiency since τ = 2.0. For the cases t = 8.0 and t = 10.0, the stall forces are λ = 0.326 and λ = 0.375, respectively. Unlike the steady
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0.8
0.000
0.6
0.002
W
(a) 0.002
Pref
(a) 1.0
0.004
0.4
0.006
0.2 0.4
0.5
0.6
0.7
0.5
0.8
1.5
2.0
2.5
3.0
3.5
4.0
14
16
18
20
Ε
Λ
(b)
(b) 1.0
1.65
0.8
1.60 1.55
Εopt
0.6
Pref
1.0
1.50
0.4
1.45
0.2
1.40 1.35
0.0 0.4
0.5
0.6
0.7
0.8
0.9
6
1.0
state case, near the stall force η = 0, showing (for small t) the engine operates irreversibly even at stall force. Far from the stall force η exhibits a pronounced peak, ηopt , at a certain optimal load, λopt . ηopt and λopt increase with t. The plot of η versus shows that the steady state efficiency (t = 30) decreases from its Carnot efficiency as increases. For small t, near the stall force η = 0. Far from the stall force η manifests a peak at certain opt . opt shifts to the left as t increases [see Fig. 5(b)]. We also evaluate Ep near the stall force and find that Ep = 0 for the steady state case only. For a large load, the temperature is not strong enough to renormalize the effect of the load and hence the velocity of the particle becomes negative as long as the load is larger than the stall force. In this case the model may act as a refrigerator and the performance of the refrigerator increases with t towards its steady state performance as a refrigerator: Pref =
( − 2λ) . 3λ
(40)
The quasistatic performance of the refrigerator also steps up with time towards its Carnot refrigerator: C Pref =
1 . τ −1
(41)
10
12
t
Ε FIG. 6. (Color online) (a) Pref as a function of λ for fixed τ = 2.0 and = 2.0. The lowest blue line, the intermediate red line, and the upper black line are plotted by fixing t = 15, t = 17, and t = 30, respectively. (b) Pref versus for fixed τ = 2.0 and λ = 0.2. The lowest blue line, the intermediate red line, and the top black line are plotted by fixing t = 8.0, t = 10.0, and t = 30.0, respectively.
8
FIG. 7. (Color online) (a) The power output W˙ as a function of for fixed τ = 2.0 and λ = 0.2. In both figures, the lowest blue line, the intermediate red line, and the upper black line are plotted by fixing t = 8.0, t = 10.0, and t = 30.0, respectively. (b) opt versus t for fixed τ = 2.0 and λ = 0.2.
Pref as a function of λ is plotted in Fig. 6(a). The figure depicts for t = 30 that the system is at steady state and that Pref monotonously decreases with λ. Near the stall force λ = 0.4, Pref = 1.0, which is the Carnot refrigerator since τ = 2.0. Far small t (t = 8.0 and t = 10.0), Pref = 0 at stall force. Pref also exhibits a pronounced peak at a certain optimal load, λopt . λopt decreases with t. The plot of Pref as a function of exhibits that Pref (t = 30) increases towards the Carnot refrigerator as increases. For small t, Pref shows a peak at a certain opt . opt shifts to the right as t increases [see Fig. 6(b)]. The power output W˙ as a function of the barrier height depicts that such a thermal engine delivers finite power while operating irreversibly. The output power increases with t and has a peak at a certain opt . This corresponds to the point in the parameter space where the engine operates with maximum power [see Fig. 7(a)]. opt also explicitly depends on t. As t steps up, opt increases and saturates to a constant value as shown in Fig. 7(b). A closer look at Fig. 7(a) once again reveals that the power output is negative (which implies a positive power input) when < 1.013, < 1.002, and < 1.0 for parameter choices of t = 8, t = 10, and t = 30, respectively. This implies that for small the power output is negative (positive power input) and the heat device works as a refrigerator [see Fig. 8(a)]. In order to clarify this, the phase diagram [Fig. 8(a)] is plotted with the
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the maximum performance of a refrigerator are quite different in optimization in the working regimes. For instance, as shown in Fig. 6(b), the performance of the refrigerator clearly exhibits an optimal value at certain opt as long as t is small. However, in the regime where the model works as a refrigerator [see Fig. 8(b)], the power input does not manifest an optimal value for any . From the above analyses one can deduce that for the engine to operate reversibly, it not only have to be in the steady state regime but also has to operate near the stall force. This implies that for small t Carnot efficiency or Carnot refrigerator is unattainable as there is irreversible heat flow from the hot to the cold reservoirs. So far we consider only the heat flow due to the change of the potential energy. If one includes the heat flow due to the kinetic energy, Carnot efficiency or Carnot refrigeration is unachievable for any parameter choice as is discussed in the next section.
(a) 4.0 3.5 3.0
Ε
2.5 2.0 1.5 1.0 0.5 5
(b)
10
15
t
20
25
30
0.00 0.02
W
0.04 0.06 0.08 0.10 0.12 0
2
4 Ε
6
8
FIG. 8. (Color online) (a) The phase space and t for parameter choices λ = 0.2 and τ = 2. The red shaded region (the upper region) depicts the regime where the model acts as a heat engine. The region where the model works as a refrigerator is marked with blue (the lower region) while the white region depicts the regime where the model acts neither as a heat engine nor as a refrigerator. (b) The power input W˙ as a function of for fixed τ = 2.0 and λ = 0.6. In the figure, the blue, red, and black lines stand for the line plotted by fixing t = 15.0, t = 17.0, and t = 30.0, respectively.
same parameter choice as in Fig. 7(a). As shown in Figs. 7(a) and 8(a), in a concrete range of values, the heat device works as a heat engine, i.e., when > 1.013, > 1.002, and > 1.0 for parameter choices of t = 8, t = 10, and t = 30, respectively. On the other hand, for small t (t = 8 and t = 10) and large values, the power output is once again negative. In this regime as shown in the phase diagram (Fig. 8), the model works neither as a heat engine nor as a refrigerator. At this point we want to emphasize that the efficiency and power output exhibit optimal values at certain opt [see Figs. 5(a) and 7(a)]. However the magnitude of opt (at which the efficiency is maximum) is quite different from that of the power output. The above analysis shows that, in general, the maximum power and the maximum efficiency as well as
V. IRREVERSIBILITY DUE TO THE HEAT FLOW VIA THE KINETIC ENERGY
In this section, we investigate the effect of heat exchange via the kinetic energy on the performance of the engine. This effect has been discussed already in Refs. [16,20,22,23]. However, most of these works address how the performance of the engine behaves in a steady state regime. Instead of studying only the long time behavior of the system, we would like to explore the model system for any time t. Our analytic result reveals that the efficiency or the performance of the refrigerator never goes to the Carnot efficiency or Carnot refrigerator at the quasistatic limit. When the particle moves from the hot to the cold heat baths, in one cycle, a 12 (τ − 1) amount of energy is transferred from the hot to the cold heat baths via kinetic energy [16,20,22,23]. Similarly, when the engine acts as a refrigerator, the net flow of the particle is from the cold to the hot reservoirs. In this case, due to particle recrossing between the hot and the cold reservoirs, heat is leaking from the hot to the cold reservoir at a magnitude of 12 (τ − 1). The rate of the heat flow (from the hot to the cold heat baths due to kinetic energy) is given ˙ ∗ (t) = 1 kB (Th − Tc )J . Hence the total heat per unit time by Q h 2 taken from the hot reservoir has the form 1 ˙ irr ˙ Q h (t) = Qh (t) + 2 (τ − 1)J,
(42)
while the total heat per unit time from the cold reservoir is given by 1 ˙ irr ˙ Q c (t) = Qc (t) + 2 (τ − 1)J.
(43)
Note that, when the engine functions as a refrigerator, Q˙ irr c (t) = ˙ c (t) − 1 (1 − τ )J . Here J = V /3. Q 2 ˙ irr (t) < η and P ∗ = Q ˙ irr The efficiency η∗ = λV /Q c (t)/ h ref λV < Pref . Before we explore the general dependence of η∗ ∗ and Pref on system parameters, let us determine the efficiency and performance of the refrigerator at steady state. At steady state
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η∗ =
6λ 2 + 2λ − 1 + τ
(44)
THERMODYNAMIC FEATURE OF A BROWNIAN HEAT . . .
PHYSICAL REVIEW E 89, 012143 (2014)
and
(a) 0.40 2 − 4λ + 1 − τ . 6λ
τ −1 In the quasistatic limit (λ → 1+2τ ), the steady state efficiency takes the form
η∗ =
τ −1 = ηC , τ
0.35
(45)
0.30
Η irr
∗ = Pref
0.20
(46)
0.15
where 6 = 2(3 + τ ) −
1 τ
−1
.
2(3 − τ 2 ) + τ + 1 . 6
(b)
0.30
0.35
0.5 0.4 0.3
(48)
where
0.25
Λ
P ref
1 ∗ C Pref
= Pref (t) =
, τ −1
0.10 0.20
(47)
Here 0 < < 1, revealing that the efficiency never approaches the Carnot efficiency ηC . Hence η∗ < η. In the ∗ converges to quasistatic limit, the steady state Pref
=
0.25
0.2 0.1
(49)
0.0
C Pref
Note that 0 < < 1, revealing that Carnot refrigeration is unattainable. We need to justify the above analysis via different approaches and retrieve the results shown above without considering the heat dissipation rate. When the engine functions as a heat engine, in one cycle, the particles takes a Q∗h = ( + λ) + 12 (τ − 1) amount of heat from the hot bath. The heat flow to the cold reservoir is given by Q∗c = ( − 2λ) + 1 (τ − 1). On the other hand when the heat engine acts as a 2 refrigerator the net heat flow out of the cold bath is given ∗ by Q∗c = ( − 2λ) − 12 (τ − 1) [20]. Evaluating η∗ and Pref , we find them to be the same as Eqs. (44) and (45). In the quasistatic limit, we retrieve Eqs. (46) and (48). Far from steady state, we evaluate the behavior of η∗ and ∗ Pref . Figure 9(a) is plotted by taking the same parameter choice as that in Fig. 5(a). However, η∗ is much less than the corresponding η, showing irreversible heat from the hot ∗ to the cold reservoir. Pref as a function of load is plotted in Fig. 9(b) by taking the same parameter choice as in Fig. 6(a). ∗ The figure depicts that Pref < Pref .
VI. SUMMARY AND CONCLUSION
In this work, we present a simple model which has few ingredients. The exact analytical results reveal the sensitivity of the performance of the thermal engine to the time t. Its operation regimes are dictated by the operation time t. In the early particle relaxation period (small t), the engine neither acts as a heat engine nor as a refrigerator. This is because, when the system relaxation time is less than the time that the engine needs to perform work, the energy taken from the hot bath dissipates without doing any work. When t further increases, depending on the parameter choice, the motor may work as a heat engine or as a refrigerator. Its performance
0.5
0.6 Λ
0.7
0.8
FIG. 9. (Color online) (a) The efficiency η as a function of λ for the same parameter choice as in Fig. 5(a). (b) Pref versus λ for the same parameter choice as in Fig. 6(a).
is also an increasing function of t. Furthermore the engine depicts a higher efficiency or performance as a refrigerator at steady state. Moreover, we show that, when one omits the heat exchange via the kinetic energy, performance as a Carnot refrigerator and Carnot efficiency are attainable only when the system operates quasistatically in the steady state regime. The magnitude and the direction of the velocity are also controlled by t. We stress that the size ratchet potential can be varied on a mesoscopic or macroscopic scale, implying that a reasonable temperature difference can be imposed along the ratchet potential [29]. The dynamics of such a system can be realized experimentally (at least for the continuum case). One makes a negatively charged monomer, then puts the particle within a positively and negatively charged fluidic channel [30]. Part of the fluidic channel is also heated. Since the Brownian particle is negatively charged, it encounters difficulty crossing through the negatively charged part of the channel. However, assisted by the nonuniform thermal background kicks, it ultimately attains a unidirectional particle current. The presence of external force may further influence the direction and the magnitude of the velocity of the particle. In conclusion, in this work we present a simple model which serves as a basic tool for better understanding of nonequilibrium statistical physics not only in the regime of NESS but also at any time t. The present model also serves as
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a tool to check many elegant thermodynamic theories. Based on this exactly solvable model, we have uncovered several thermodynamic relations.
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ACKNOWLEDGMENT
I would like to thank Y. Shiferaw and M. Bekele for the interesting discussions.
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