Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems 19 Heuristically, the underlying time-inhomogeneous Markov process D(t) can be conceived as an ensemble of individual realizations (sample paths). A realization is specified by a succession of transitions between the two states. If we know the number n of the transitions during a path and the times t k n k =1 at which they occur, we can calculate the probability that this specific path will be generated. A given paths yields a unique value of the microscopic work done on the system. For example, if the system is known to remain during the time interval [t k ,t k+1 ] in the ith state, the work done on the system during this time interval is simply E i (t k+1 ) −E i (t k ). The probability of an arbitrary fixed path amounts, at the same time, the probability of that value of the work which is attributed to the path in question. Viewed in this way, the work itself is a stochastic process and we denote it as W (t). We are interested in its probability density ρ (w, t)=δ(W(t) − w),where  denotes the average over all possible paths. We now introduce the augmented process { W(t), D(t) } which simultaneously reflects both the work variable and the state variable. The augmented process is again a time non-homogeneous Markov process. Actually, if we know at a fixed time t  both the present state variable j and the work variable w  , then the subsequent probabilistic evolution of the state and the work is completely determined. The work done during the time period [t  ,t], where t > t  , simply adds to the present work w  and it only depends on the succession of the states after the time t  . And this succession by itself cannot depend on the dynamics before time t  . The one-time properties of the augmented process will be described by the functions G ij (w, t |w  ,t  )=lim →0 Prob { W(t) ∈ (w, w + ) and D(t)=i |W (t  )=w  and D(t  )=j }  , (50) where i, j = 1, 2. We represent them as the matrix elements of a single two-by-two matrix G (w, t |w  ,t  ), G ij (w, t |w  ,t  )=i |G(w, t |w  ,t  ) |j  . (51) We need an equation which controls the time dependence of the propagator G (w, t |w  ,t  ) and which plays the same role as the Master equation (43) in the case of the simple two-state process. This equation reads (Imparato & Peliti, 2005b; ˇ Subrt & Chvosta, 2007) ∂ ∂t G (w, t |w  ,t  )=−  dE 1 (t) dt 0 0 dE 2 (t) dt  ∂ ∂w +  λ 1 (t) −λ 2 (t) − λ 1 (t) λ 2 (t)   G (w, t |w  ,t  ), (52) where the initial condition is G (w, t  |w  ,t  )=δ(w − w  )I. The matrix equation represents a hyperbolic system of four coupled partial differential equations with the time-dependent coefficients. Similar reasoning holds for the random variable Q(t) which represents the heat accepted by the system from the environment. Concretely, if the system undergoes during a time interval [t k ,t k+1 ] only one transition which brings it at an instant τ ∈ [t k ,t k+1 ] from the state i to the state j, the heat accepted by the system during this time interval is E j (τ) − E i (τ).Thevariable Q(t) is described by the propagator K(q, t |q  ,t  ) with the matrix elements K ij (q, t |q  ,t  )=lim →0 Prob  Q(t) ∈ (q, q + ) ∧ D( t)=i |Q(t  )=q  ∧ D(t  )=j   . (53) 171 Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems 20 Thermodynamics It turns out that there exists a simple relation between the heat propagator and the work propagator G (w, t |w  ,t  ). Since for each path, heat q and work w are connected by the first law of thermodynamics, we have q = E i (t) − E j (t  ) −w for any path which has started at the time t  in the state j and which has been found at the time t in the state i. Accordingly, K (q, t |q  ,t  )=  g 11 (u 11 (t, t  ) −q, t |q  ,t  ) g 12 (u 12 (t, t  ) −q,t |q  ,t  ) g 21 (u 21 (t, t  ) −q, t |q  ,t  ) g 22 (u 22 (t, t  ) −q,t |q  ,t  )  , (54) where u ij (t, t  )=E i (t) − E j (t  ). The explicit form of the matrix G (w, t) which solves the dynamical equation (52) with the Glauber transition rates (45) and the periodically modulated energies (46) can be found in (Chvosta et al., 2010). Heaving the matrix G (w, t) for the limit cycle, the matrix K(q, t) is calculated using the transformation (54). In the last step, we take into account the initial condition |π at the beginning of the limit cycle and we sum over the final states of the process D (t). Then the (unconditioned) probability density for the work done on the system in the course of the limit cycle reads ρ (w, t)= 2 ∑ i=1 i |G(w,t)|π  . (55) Similarly, the probability density for the heat accepted during the time interval [0, t] is χ (q, t)= 2 ∑ i=1 i |K(q, t)|π . (56) The form of the resulting probability densities and therefore also the overall properties of the engine critically depend on the two dimensionless parameters a ± = νt ± /(2β ± |h 2 − h 1 |). We call them reversibility parameters 1 . For a given branch, say the first one, the parameter a + represents the ratio of two characteristic time scales. The first one, 1/ν, describes the attempt rate of the internal transitions. The second scale is proportional to the reciprocal driving velocity. Contrary to the first scale, the second one is fully under the external control. Moreover, the reversibility parameter a + is proportional to the absolute temperature of the heat bath, k B /β + . F IG. 7 illustrates the shape of the limit cycle together with the functions ρ(w, t p ), χ( q, t p ) for various values of the reversibility parameters. Notice that the both functions ρ (w, t p ) and χ (q, t p ) vanishes outside a finite support. Within their supports, they exhibit a continuous part, depicted by the full curve, and a singular part, illustrated by the full arrow. The height of the full arrow depicts the weight of the corresponding δ-function. The continuous part of the function ρ (w, t p ) develops one discontinuity which is situated at the position of the full arrow. Similarly, the continuous part of the function χ (q, t p ) develops three discontinuities. If the both reversibility parameters a ± are small, the isothermal processes during the both branches strongly differ from the equilibrium ones. The indication of this case is a flat continuous component of the density ρ (w, t p ) and a well pronounced singular part. The strongly irreversible dynamics occurs if one or more of the following conditions hold. First, if ν is small, the transitions are rare and the occupation probabilities of the individual energy 1 The reversibility here refers to the individual branches. As pointed out above, the abrupt change in temperature, when switching between the branches, implies that there exists no reversible limit for the complete cycle. 172 Thermodynamics Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems 21 0 2 4 6 −1 −0.5 0 p(t) −8 −4 0 4 8 0 0.2 0.4 W (t p ) −10 −5 0 5 10 0 0.2 0.4 Q(t p ) χ(q, t p )[J −1 ] a) 0 2 4 6 −1 −0.5 0 −8 −4 0 4 8 0 0.2 0.4 W (t p ) −10 −5 0 5 10 0 0.2 0.4 Q(t p ) b) 0 2 4 6 −1 −0.5 0 −8 −4 0 4 8 0 0.2 0.4 W (t p ) −10 −5 0 5 10 0 0.2 0.4 Q(t p ) c) 0 2 4 6 −1 −0.5 0 E(t)[J] −8 −4 0 4 8 0 0.2 0.4 w [J] ρ(w, t p )[J −1 ] W (t p ) −10 −5 0 5 10 0 0.2 0.4 Q(t p ) d) q [J] Fig. 7. Probability densities ρ(w, t p ) and χ(q, t p ) for the work and the heat for four representative sets of the engine parameters (every set of parameters corresponds to one horizontal triplet of the panels). The first panel in the triplet shows the limit cycle in the p −E plane (p (t)=p 1 (t) − p 2 (t) is the occupation difference and E(t)=E 1 (t)). In the parametric plot we have included also the equilibrium isotherm which corresponds to the first stroke (the dashed line) and to the second stroke (the dot-dashed line). In all panels we take h 1 = 1J,h 2 = 5J,andν = 1s −1 . The other parameters are the following. a in the first triplet: t + = 50 s, t − = 10 s, β + = 0.5 J −1 , β − = 0.1 J −1 , a ± = 12.5 (the bath of the first stroke is colder than that of the second stroke). b in the second triplet: t + = 50 s, t − = 10 s, β + = 0.1 J −1 , β − = 0.5 J −1 , a + = 62.5, a − = 2.5 (exchange of β + and β − as compared to case a, leading to a change of the traversing of the cycle from counter-clockwise to clockwise and a sign reversal of the mean values W (t p ) ≡W(t p )  and Q(t p ) ≡Q(t p )  ). c in the third triplet: t + = 2s, t − = 2s,β + = 0.2 J −1 , β − = 0.1 J −1 , a + = 1.25, a − = 2.5 (a strongly irreversible cycle traversed clockwise with positive work). d in the fourth triplet: t + = 20 s, t − = 1s, β ± = 0.1 J −1 , a + = 25, a − = 1.25 (no change in temperatures, but large difference in duration of the two strokes; W (t p ) is necessarily positive). The height of the red arrows plotted in the panels with probability densities depicts the weight of the corresponding δ-functions. levels are effectively frozen during long periods of time. Therefore they lag behind the Boltzmann distribution which would correspond to the instantaneous positions of the energy levels. More precisely, the population of the ascending (descending) energy level is larger (smaller) than it would be during the corresponding reversible process. As a result, the mean work done on the system is necessarily larger than the equilibrium work. Secondly, a similar situation occurs for large driving velocities v ± . Due to the rapid motion of the energy levels, the occupation probabilities again lag behind the equilibrium ones. Thirdly, the strong irreversibility occurs also in the low temperature limit. In the limit a ± → 0, the continuous part vanishes and ρ (w, t p )=δ(w). In the opposite case of large reversibility parameters a ± , the both branches in the p −E plane are located close to the reversible isotherms. The singular part of the density ρ (w, t p ) is suppressed and the continuous part exhibits a well pronounced peak. The density ρ (w, t p ) approaches the Gaussian function centered around the men work. This confirms the general 173 Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems 22 Thermodynamics considerations (Speck & Seifert, 2004). In the limit a ± → ∞ the Gaussian peak collapses to the delta function located at the quasi-static work (Chvosta et al., 2010). The heat probability density χ (q, t p ) shows similar properties as ρ(w, t p ). 6. 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Introduction In most introductory physics texts, a discussion on (human) food consumption centers around the available work. For example, the altitude is calculated a person can hike after eating a snack. This connection is natural at first glance: food is burned in a bomb calorimeter and its energy content is measured in “Calories,” which is the unit of heat. We say that “we go to the gym to burn calories.” This discussion implies that the human body acts as a sort of “heat engine,” with food playing the role of ‘fuel.’ We give two arguments to show that this view is flawed. First, the conversion of heat into work requires a heat engine that operates between two heat baths with different temperatures T h and T c < T h . The heat input Q h can be converted into work W and heat output Q c < Q h so that Q h = Q c + W subject to the condition that entropy cannot be destroyed: ΔS = Q h /T h − Q c /T c > 0. However, animals act like thermostats, with their body temperature kept at a constant value; e.g., 37 ◦ C for humans and 1 −2 ◦ C higher for domestic cats and dogs. Second, the typical diet of an adult is roughly 2,000 Calories or about 8 MJ. If we assume that 25% of caloric intake is converted into useable work, a 100-kg adult would have to climb about 2,000 m [or approximately the height of Matterhorn in the Swiss Alps from its base] to convert daily food intake into potential energy. While this calculation is too simplistic, it illustrates that caloric intake through food consumption is enormous, compared to mechanical work done by humans [and other animals]. In particular, the discussion ignores heat production of the skin. At rest, the rate of heat production per unit area is F/A  45W/m 2 (Guyton & Hall 2005). Given that the surface area of a 1.8-m tall man is about A 2m 2 , the rate of energy conversion at rest is approximately 90 W. Since 1d  9 × 10 4 s, we find that the heat dissipated through the skin is F8MJ/d, which approximately matches the daily intake of ‘food calories.’ An entirely different focus of food consumption is emphasized in physiology texts. All living systems require the input of energy, whether it is in the form of food (for animals) or sun light (for plants). The chemical energy content of food is used to maintain concentration gradients of ions in the body, which is required for muscles to do useable work both inside and outside the body. Heat is the product of this energy transformation. That is, food intake is in the form of Gibbs free energy, i.e., work, and entropy is created in the form of heat and other waste products. In his classic text What is Life?, Schr ¨ odinger coined the expression that living systems “feed on negentropy” (Schr ¨ odinger 1967). Later, Morowitz explained that the steady state of living systems is maintained by a constant flow of energy: the input is highly organized energy [work], while the output is in the form of disorganized energy, and entropy 9 2 Thermodynamics is produced. Indeed, energy flow has been identified as one of the principles governing all complex systems (Schneider & Sagan 2005). As an example of the steady-state character of living systems with non-zero-gradients, we discuss the distribution of ions inside the axon and extracellular fluid. The ionic concentrations inside the axon c i and in the extracellular fluid c o are measured in units of millimoles per liter (Hobbie & Roth 2007): Ion c i c o c o /c i Na + 15 145 9.7 K + 150 5 0.0033 Cl − 9 125 13.9 Misc. − 150 30 0.19 In thermal equilibrium, the concentration of ions across a cell membrane is determined by the Boltzmann-Nernst formula, c i /c o = exp[−ze(v i − v o )/k B T] , where ΔG = ze(v i − v o ) is the Gibbs free energy for the potential between the inside and outside the cell, Δv = v i − v o . If the electrostatic potential in the extracellular fluid is chosen v o = 0, the ‘resting’ potential inside the axon is found v i = −70mV. For T = 37 ◦ C, this gives c i /c o = 13.7 and c i /c o = 1/13.7 = 0.073 for univalent positive and negative ions, respectively. That is, the sodium concentration is too low inside the axon, while there are too many potassium ions inside it. The concentration of chlorine is approximately consistent with thermal equilibrium. Non-zero gradients of concentrations and other state variables are characteristic for systems that are not in thermal equilibrium (Berry et. al. 2002). A discussion of living and complex systems within the framework of physics is difficult. It must include an explanation of what is meant by the phrase “biological systems are in nonequilibrium stationary states (NESS).” This is challenging, because there is not a unique definition of ’equilibrium state;’ rather entirely different definitions are used to describe closed and open systems. For a closed system, the equilibrium state can be characrterized by a (multi-dimensional) coordinate x s , so that x = x s describes a nonequilibrium state. However, the notion of “state of the system” is far from obvious for open systems. For a population model in ecology, equilibrium is described by the number of animals in each species. A nonequilibrium state involves populations that are changing with time, so a ‘nonequilibrium stationary state’ would correspond to dynamic state with constant (positive or negative) growth rates for species. Thus, any discussion of nonequilibrium thermodynamics for biological systems must involve an explanation of ‘state’ for complex systems. For many-body systems, the macroscopic behavior is an “emergent behavior;” the closest analogue of ‘state’ in physics might be the order parameter associated with a broken symmetry near a second-order phase transition. This chapter is not a comprehensive overview of nonequilibrium thermodynamics, or the flow of energy as a mechanism of pattern formation in complex systems. We begin by directing the reader to some of the texts and papers that were useful in the preparation of this chapter. The text by de Groot and Mazur remains an authoritative source for nonequilibrium thermodynamics (de Groot & Mazur 1962). Applications in biophysics are discussed in Ref. (Katchalsky & Curram 1965). The text by Haynie is an excellent introduction to biological thermodynamics (Haynie 2001). The texts by Kubo and coworkers are an authoritative treatment of equilibrium and nonequilibrium statistical mechanics (Toda et al 1983; Kubo et al 1983). Stochastic processes are discussed in Refs. (Wax 1954; van Kampen 1981). Sethna gives a clear explanation of complexity and entropy 178 Thermodynamics Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description 3 (Sethna 2006). Cross and Greenside overview pattern formation in dissipative systems (Cross & Greenside 2009); a non-technical introduction to pattern formation is found in Ref. (Ball 2009). The reader is directed to Refs. (Guyton & Hall 2005) and (Nobel 1999) for background material on human and plant physiology. Some of the physics underlying human physiology is found in Refs. (Hobbie & Roth 2007; Herman 2007). The outline of this paper is as follows. We discuss the meaning of state and equilibrium for closed systems. We then discuss open systems, and introduce the concept of order parameter as the generalization of “coordinate” for closed systems. We use the motion of a Brownian particle to illustrate the two mechanisms, namely fluctuation and dissipation, how a system interacts with a much larger heat bath. We then briefly discuss the Rayleigh-Benard convection cell to illustrate the nonequilibrium stationary states in dissipative systems. This leads to our treatment of a charged object moving inside a viscous fluid. We discuss how the flow of energy through the system determines the stability of NESS. In particular, we show how the NESS becomes unstable through a seemingly small change in the energy dissipation. We conclude with a discussion of the key points and a general overview. 2. Closed systems The notion of ‘equilibrium’ is introduced for mechanical systems, such as the familiar mass-block system. The mass M slides on a horizontal surface, and is attached to a spring with constant k, cf. Fig. (1). We choose a coordinate such that x eq = 0 when the spring force vanishes. The potential energy is then given by U (x)=kx 2 /2, so that the spring force is given by F sp (x)=−dU/dx = −kx. In Fig (1), the potential energy U(x) is shown in black. If the coordinate is constant, x ns = const = 0, the spring-block system is in a nonequilibrium stationary state. Since F sp = −dU/dx | ns = 0, an external force must be applied to maintain the system in a steady state: F net = F sp + F ext = 0. If the object with mass M also has an electric charge q, this external force can be realized by an external electric field E, F ext = qE. The external force can be derived from a potential energy F ext = −dU ext /dx with U ext = −qEx, and the spring-block system can be enlarged to include the electric field. Mathematically. this is expressed in terms of a total potential energy that incorporates the interaction with the electric field: U → U  = U + U ext , where U  (x)= 1 2 kx 2 − F ext x = 1 2 k ( x − x ns ) 2 − ( qE) 2 4k . (1) The potential U  (x) is shown in red. That is, the nonequilibrium state for the potential U(x), x ns corresponds to the equilibrium state for the potential U  (x), x  s : x ns = x  s = qE k . (2) That is, the nonequilibrium stationary state for the spring-block system is the equilibrium state for the enlarged system. We conclude that for closed systems, the notion of equilibrium and nonequilibrium is more a matter of choice than a fundamental difference between them. For a closed system, the signature of stability is the oscillatory dynamics around the equilibrium state. Stability follows if the angular frequency ω is real: ω 2 = 1 M d 2 U dx 2 > 0 (stability), (3) 179 Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description 4 Thermodynamics That is, stability requires that the potential energy is a convex function. Since d 2 U/dx 2 = d 2 U/dx 2 , the stability of the system is not affected by the inclusion of the external electric force. On the other hand, if the angular frequency is imaginary ω = i ˜ ω, such that d 2 U dx 2 < 0 (instability). (4) The corresponding potential energy is shown in Fig. (2). The solution of the equation of motion describes exponential growth. That is, a small disturbance from the stationary state is amplified by the force that drives the system towards smaller values of the potential energy for all initial deviations from the stationary state x = 0, x (t) −→ ±∞, t −→ ∞; (5) the system is dynamically unstable. We conclude that a concave potential energy is the condition for instability in closed systems. 3. Equilibrium thermodynamics Open systems exchange energy (and possibly volume and particles) with a heat bath at a fixed temperature T. The minimum energy principle applies to the internal energy of the system, rather than to the potential energy. This principle states that “the equilibrium value of any constrained external parameter is such as to minimize the energy for the given value of the total energy” (Callen 1960). A thermodynamic description is based on entropy, which is a concave function of (constrained) equilibrium states. In thermal equilibrium, the extensive parameters assume value, such that the entropy of the system is maximized. This statement is referred to as maximum entropy principle [MEP]. The stability of thermodynamic equilibrium follows from the concavity of the entropy, d 2 S < 0. Thermodynamics describes average values, while fluctuations are described by equilibrium statistical mechanics. The distribution of the energy is given by the Boltzmann factor p (E)=Z −1 exp(−E/k B T), where Z =  exp(−E/k B T)dE is the partition function. The equilibrium value of the energy of the system is equal to the average value, E eq =  E  =  dE p(E)E. The fluctuations of the energy are δE = E −  E  . The mean-square fluctuations can be written  [δE] 2  = k B T 2 · d  E  /dT, or in terms of the inverse temperature β = 1/T,  [δE] 2  = −k B d  E  /dβ. Thus, the variance of energy fluctuations  [δE] 2  is proportional to the response of the systems d  E  /dβ. The proportionality between fluctuations and dissipation is determined by the Boltzmann constant k B = 1.38 ×10 −23 J/K. Einstein discussed that “the absolute constant k B (therefore) determines the thermal stability of the system. The relationship just found is particularly interesting because it no longer contains any quantity that calls to mind the assumption underlying the theory” (Klein 1967). In general, the state of an open system is described by an order parameter η. This concept is the generalization of coordinates used for closed systems, and was introduced by Landau to describe the properties of a system near a second-order phase transition (Landau & Lifshitz 1959a). For the Ising spin model, for example, the order parameter is the average the average magnetization (Chaikin & Lubensky 1995). In general, the choice of order parameter for a particular system is an “art” (Sethna 2006). For simplicity, we assume a spatially homogenous system, so that η (  x)=const and there is no term involving the gradient ∇η. The order parameter can be chosen such that η = 0in the symmetric phase. The thermodynamics of the system is defined by the Gibbs free energy 180 Thermodynamics [...]... stochastic non-equilibrium dynamics in terms of Fokker-Planck equations To set the groundwork for the development of the formalism, we discuss first the basic concepts of mesoscopic non-equilibrium thermodynamics and proceed afterwards with the application of the theory to non-equilibrium radiative transfer at the nanoscale 2 Mesoscopic non-equilibrium thermodynamics Mesosocopic non-equilibrium thermodynamics. .. 20 07] Callendar, C (20 07) Not so cool Metascience 16, 1 47- 151 [Camazine et al 2001] Camazine, S.; Deneubourg, J.-L; Franks, N R.; Sneyd, J.; Theraulaz, G & Bonabeau, E (2001) Self-Organization in Biological Systems (Princeton University Nonequilibrium Thermodynamics Living Systems: Brownian Particle Description Nonequilibrium Thermodynamics for for Living Systems: Brownian Particle Description 11 1 87. .. to discuss important topics in nonequilibrium thermodynamics, such as pattern formation in driven-diffusive systems Nonequilibrium Thermodynamics Living Systems: Brownian Particle Description Nonequilibrium Thermodynamics for for Living Systems: Brownian Particle Description 7 183 5 Nonequilibrium stationary states: brownian particle model Our system is a particle with mass M and the “state” of the... Gibbs Free Energy Thermodynamics Thermodynamics stripes hexagonal state Fig 4 The Gibbs free energy for the Rayleigh-Bernard connection cell F,W vs v Fig 5 The rate of energy input and energy dissipation for the Brownian particle immersed in a fluid Nonequilibrium Thermodynamics Living Systems: Brownian Particle Description Nonequilibrium Thermodynamics for for Living Systems: Brownian Particle Description... description aimed at obtaining a simple and comprehensive explanation of the dynamics of non-equilibrium systems at the mesoscopic scale The theory, mesoscopic non-equilibrium thermodynamics, has provided a deeper understanding of the concept of local equilibrium and a framework, reminiscent of non-equilibrium thermodynamics, through which fluctuations in non-linear systems can be studied The probabilistic... 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(2005) Non-equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond (Springer-Verlag, New York) [Klein 19 67] A Einstein as quoted in Klein, M J (19 67) Thermodynamics in Einstein’s Thought Science 1 57, 509-516 [Kubo et al 1983] R Kubo, M Toda and N Hashitsume (1983) Statistical Physics II:Nonequilibrium... 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Phys. 29: 305–3 07. 176 Thermodynamics 0 Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description Ulrich Z ¨ urcher Physics Department, Cleveland State University Cleveland,. Soc. Jpn. 66: 1234–12 37. Sekimoto, K., Takagi, F. & Hondou, T. (2000). Carnot’s cycle for small systems: Irreversibility and cost of operations, Phys. Rev. E 62(6): 77 59 77 68. Speck, T. &. 0, the ‘resting’ potential inside the axon is found v i = 70 mV. For T = 37 ◦ C, this gives c i /c o = 13 .7 and c i /c o = 1/13 .7 = 0. 073 for univalent positive and negative ions, respectively.