Thermodynamics 2012 Part 7 pptx

Later, Morowitz explained that thesteady state of living systems is maintained by a constant flow of energy: the input is highlyorganized energy [work], while the output is in the form of

Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems 19

Heuristically, the underlying time-inhomogeneous Markov processD(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=1at which they occur, we can calculate the probability thatthis specific path will be generated A given paths yields a unique value of the microscopicwork done on the system For example, if the system is known to remain during the timeinterval[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 asW(t) We are interested

in its probability densityρ(w, t) = δ(W(t) −w), where .denotes the average over allpossible paths

We now introduce the augmented process {W(t),D(t)} which simultaneously reflects boththe 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 tboth the present

state variable j and the work variable w, then the subsequent probabilistic evolution of thestate and the work is completely determined The work done during the time period[t, t], where t>t, simply adds to the present work wand it only depends on the succession of the

states after the time t And this succession by itself cannot depend on the dynamics before

∂

∂tG(w, t|w, t) = − d E dt1(t) 0

0 d E2(t) dt

a hyperbolic system of four coupled partial differential equations with the time-dependentcoefficients

Similar reasoning holds for the random variableQ(t)which represents the heat accepted bythe 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 instantτ∈ [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) The variable

Q(t)is described by the propagatorK(q, t|q, t)with the matrix elements

K ij(q, t|q, t) =lim

→0ProbQ(t) ∈ (q, q+) ∧D(t) =i|Q(t) =q∧D(t) =j

171Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems

It turns out that there exists a simple relation between the heat propagator and the workpropagatorG(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 tin the state j and which has been found at the time t in the state i Accordingly,

In the last step, we take into account the initial condition|πat the beginning of the limit cycleand we sum over the final states of the processD(t) Then the (unconditioned) probabilitydensity for the work done on the system in the course of the limit cycle reads

ρ(w, t) =∑2

i=1i|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=1i|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±|h2−h1|)

We call them reversibility parameters1 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 reciprocaldriving 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

of the full arrow depicts the weight of the correspondingδ-function The continuous part of

the functionρ(w, tp)develops one discontinuity which is situated at the position of the fullarrow Similarly, the continuous part of the functionχ(q, tp)develops three discontinuities

If the both reversibility parameters a± are small, the isothermal processes during the bothbranches strongly differ from the equilibrium ones The indication of this case is a flatcontinuous component of the densityρ(w, tp) and a well pronounced singular part Thestrongly 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.

Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems 21

W (t p

0.2 0.4

W (t p

0.2 0.4

W (t p

0.2 0.4

Q(t p

d)

q [J]

Fig 7 Probability densitiesρ(w, tp)andχ(q, tp)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) =p1(t) −p2(t)is the occupation difference and E( t) =E1(t)) In the parametricplot 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

h1=1 J, h2=5 J, andν=1 s−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 ofβ+andβ−as compared to case a, leading to achange of the traversing of the cycle from counter-clockwise to clockwise and a sign reversal

of the mean values W( tp) ≡ W(tp)and Q( tp) ≡ Q(tp)) c in the third triplet: t+=2 s,

t−=2 s,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−=1 s,

β±=0.1 J−1, a+=25, a−=1.25 (no change in temperatures, but large difference in duration

of the two strokes; W( tp)is necessarily positive) The height of the red arrows plotted in thepanels with probability densities depicts the weight of the correspondingδ-functions.

levels are effectively frozen during long periods of time Therefore they lag behind theBoltzmann distribution which would correspond to the instantaneous positions of the energylevels 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, themean 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 energylevels, 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 continuouspart vanishes andρ(w, tp) =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, tp) issuppressed and the continuous part exhibits a well pronounced peak The densityρ(w, tp)approaches the Gaussian function centered around the men work This confirms the general

173Four Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems

considerations (Speck & Seifert, 2004) In the limit a±→∞ the Gaussian peak collapses tothe delta function located at the quasi-static work (Chvosta et al., 2010) The heat probabilitydensityχ(q, tp)shows similar properties asρ(w, tp).

6 Acknowledgements

Support of this work by the Ministry of Education of the Czech Republic (project No MSM0021620835), by the Grant Agency of the Charles University (grant No 143610) and by theprojects SVV – 2010 – 261 301, SVV – 2010 – 261 305 of the Charles University in Prague isgratefully acknowledged

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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 tothe gym to burn calories.” This discussion implies that the human body acts as a sort of “heatengine,” with food playing the role of ‘fuel.’

We give two arguments to show that this view is flawed First, the conversion of heat intowork 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 hso

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 aconstant 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 thatcaloric intake through food consumption is enormous, compared to mechanical work done byhumans [and other animals] In particular, the discussion ignores heat production of the skin

At rest, the rate of heat production per unit area isF /A 45 W/m2 (Guyton & Hall 2005)

Given that the surface area of a 1.8-m tall man is about A 2 m2, the rate of energy conversion

at rest is approximately 90 W Since 1 d9×104s, we find that the heat dissipated throughthe skin isF 8 MJ/d, which approximately matches the daily intake of ‘food calories.’

An entirely different focus of food consumption is emphasized in physiology texts All livingsystems require the input of energy, whether it is in the form of food (for animals) or sunlight (for plants) The chemical energy content of food is used to maintain concentrationgradients of ions in the body, which is required for muscles to do useable work both insideand 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 thesteady state of living systems is maintained by a constant flow of energy: the input is highlyorganized energy [work], while the output is in the form of disorganized energy, and entropy

9

is produced Indeed, energy flow has been identified as one of the principles governing allcomplex 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 oare measured in units ofmillimoles per liter (Hobbie & Roth 2007):

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)/kB T], whereΔG=ze(v i − v o) isthe 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 = −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 That is, the sodiumconcentration 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-zerogradients 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 innonequilibrium stationary states (NESS).” This is challenging, because there is not a uniquedefinition of ’equilibrium state;’ rather entirely different definitions are used to describe closedand open systems For a closed system, the equilibrium state can be characrterized by a

(multi-dimensional) coordinate xs, so that x = xsdescribes a nonequilibrium state However,the notion of “state of the system” is far from obvious for open systems For a populationmodel in ecology, equilibrium is described by the number of animals in each species Anonequilibrium state involves populations that are changing with time, so a ‘nonequilibriumstationary state’ would correspond to dynamic state with constant (positive or negative)growth rates for species Thus, any discussion of nonequilibrium thermodynamics forbiological systems must involve an explanation of ‘state’ for complex systems For many-bodysystems, the macroscopic behavior is an “emergent behavior;” the closest analogue of ‘state’ inphysics might be the order parameter associated with a broken symmetry near a second-orderphase transition

This chapter is not a comprehensive overview of nonequilibrium thermodynamics, orthe flow of energy as a mechanism of pattern formation in complex systems Webegin by directing the reader to some of the texts and papers that were useful in thepreparation of this chapter The text by de Groot and Mazur remains an authoritativesource for nonequilibrium thermodynamics (de Groot & Mazur 1962) Applications inbiophysics are discussed in Ref (Katchalsky & Curram 1965) The text by Haynie is anexcellent introduction to biological thermodynamics (Haynie 2001) The texts by Kuboand 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

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) forbackground material on human and plant physiology Some of the physics underlying humanphysiology is found in Refs (Hobbie & Roth 2007; Herman 2007)

The outline of this paper is as follows We discuss the meaning of state and equilibriumfor closed systems We then discuss open systems, and introduce the concept of orderparameter as the generalization of “coordinate” for closed systems We use the motion of aBrownian 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-Benardconvection cell to illustrate the nonequilibrium stationary states in dissipative systems Thisleads to our treatment of a charged object moving inside a viscous fluid We discuss how theflow of energy through the system determines the stability of NESS In particular, we showhow 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 xeq=0 when the spring force

vanishes The potential energy is then given by U( x) =kx2/2, so that the spring force is given

by Fsp( x ) = − dU/dx = − kx In Fig (1), the potential energy U(x)is shown in black

If the coordinate is constant, xns=const= 0, the spring-block system is in a nonequilibrium stationary state Since Fsp= − dU/dx |ns=0, an external force must be applied to maintain the

system in a steady state: Fnet=Fsp+Fext=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, Fext=qE.

The external force can be derived from a potential energy Fext = − dUext/dx with Uext= − 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+Uext, where

The potential U (x)is shown in red That is, the nonequilibrium state for the potential U( x),

xnscorresponds to the equilibrium state for the potential U (x), x

For a closed system, the signature of stability is the oscillatory dynamics around the

equilibrium state Stability follows if the angular frequencyω is real:

That is, stability requires that the potential energy is a convex function Since d2U/dx2=

d2U/dx2, the stability of the system is not affected by the inclusion of the external electricforce On the other hand, if the angular frequency is imaginaryω=i ˜ ω, such that

d2U

The corresponding potential energy is shown in Fig (2) The solution of the equation ofmotion describes exponential growth That is, a small disturbance from the stationary state isamplified by the force that drives the system towards smaller values of the potential energy

for all initial deviations from the stationary state x=0,

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 anyconstrained external parameter is such as to minimize the energy for the given value of thetotal energy” (Callen 1960) A thermodynamic description is based on entropy, which is aconcave 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, d2S <0

Thermodynamics describes average values, while fluctuations are described by equilibriumstatistical mechanics The distribution of the energy is given by the Boltzmann factor

p(E) =Z −1exp(−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, Eeq =  E  =

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 theaverage the average magnetization (Chaikin & Lubensky 1995) In general, the choice of orderparameter for a particular system is an “art” (Sethna 2006)

For simplicity, we assume a spatially homogenous system, so thatη ( x) =const and there is

no term involving the gradient∇ η The order parameter can be chosen such that η=0 inthe symmetric phase The thermodynamics of the system is defined by the Gibbs free energy

Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description 5

G=G(η) The equation of state follows from the expression for the external field h=∂G/∂η.

The susceptibilityχ=∂  η /∂h characterizes the response of the system In the absence of an

external magnetic field, the appearance of a non-zero value of the order parameter is referred

to as spontaneous symmetry breaking (Chaikin & Lubensky 1995; Forster 1975) The Gibbs free energy is written as a power series G( η) =G0+Aη2+Bη4 with B >0 The symmetricphaseη=0, corresponds to A > 0, while A <0 in the asymmetric phase The second-order

coefficient vanishes at the transition point A=0 We only consider the case when the system

is away from the transition point, so that A =0, and write ¯η=0 and ¯η = ± A/2B for the

respective minima of the Gibbs free energy, respectively Since ∂2G/∂η2

η = ¯η=χ −1 >0, thesusceptibility is finite and the variance of fluctuations of the order parameter is finite as well,with

[η − η¯]2

∼ χk B T, which is referred to as fluctuation-dissipation theorem [FDT].

A Brownian particle can be used to illustrate some aspects of equilibrium statistical mechanics

(Forster 1975) In a microscopic description, a heavy particle with mass M is immersed in a fluid of lighter particles of mass m < M The time evolution is described by the Liouville

operator for the entire system, and projection operator methods are used to eliminate thelighter particles’ degrees of freedom [i.e., the heat bath] It is shown that the interactionwith a heat bath results in dissipation, described by a memory function and fluctuationscharacterized by stochastic forces Because these two contributions have a common origin,

it is not surprising that they are related to each other: the memory function is proportional tothe autocorrelation function of the stochastic forces The average kinetic energy of the heavyparticle is given by the equipartition principle: (M/2)v2

=k B T/2 The memory function

defines a correlation timeζ −1 For times t > ζ −1, a Langevin equation for the velocity of the

heavy particle follows (Wax 1954) In one spatial dimension,

fact, following Kubo, Eq (7) is sometimes called the ‘second fluctuation-dissipation theorem.’

For long times, t >> ζ −1 , the mean-square displacement increases diffusively:

[x(t ) − x(0)]2

=2k B T

The expression for the diffusion constant D=k B T/Mζ is the Einstein relation for Brownian

motion, and is a version of the fluctuation-dissipation theorem The diffusion constant can be

written in terms of the velocity autocorrelation, D=0∞dt  v(t ) v(0)

If the Brownian particle moves in a harmonic potential well, U( x) =Mω2x2/2, the Langevin

equation is written as a system of two coupled first-order differential equations: dx/dt=v and dv/dt+ζv+ω2x=ζ/M If the damping constant is large, the inertia of the particle can

be ignored, so that the coordinate is described by the equation: dx/dt+ (ω2/ζ)x=ζ/M At

zero temperature, the stochastic force vanishes, and the deterministic time evolution of the

coordinate describes its relaxation towards the equilibrium x=0: dx/dt = −( ω2/ζ)x so that

x(t) =x0exp[−(ω2/ζ)t]

In general, the relaxation of an initial nonoequilibrium state is governed by Onsager’sregression hypothesis: the decay of an initial nonequilibrium state follows the same law as that

181Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description

of spontaneous fluctuations (Kubo et al 1983) The fluctuation-dissipation theorem implies

that the time-dependence of equilibrium fluctuations is governed by the minimum entropyproduction principle The stochastic nature of time-depedent equilibrium fluctuations is

characterized by the conditional probability, or propagator P( x, t | x0, t0). Onsager andMachlup showed that the conditional probability, or propagator, for diffusion can be written

as a path integral (Hunt et al 1985):

P(x, t | x0, t0) = D[ x(t)]exp

− ζ 2k B T

subject to the initial condtion P( x, t0| x0, t0) =δ(x − x0).

4 Systems far from equilibrium

We conclude that dissipation tends to ‘dampen’ the oscillatory motion around the equilibriumvalue ¯η, so that lim t→∞  η(t ) = η Thus, a nonequilibrium stationary state η¯ s= η requires the¯input of energy through work done on the system: highly-organized energy is destroyed, anddissipated energy is associated with the production of heat

This mechanism is often illustrated by the Rayleigh-Benard convection cell, with a fluid beingplaced between two horizontal plates If the two plates are at the same temperature, there

is no macroscopic fluid flow, and the system is in the symmetric phase An energy input isused to maintain a constant temperature difference across the plates IfΔT is large enough, the component of the velocity along the vertical is non-zero, v z = 0 A state with v z =0 is theasymmetric state of the fluid Stationary patterns such as “stripes” and “hexagons” developinside the fluid Thus, the temperature differenceΔT can be viewed as the “force” maintaining stationary patterns in the fluid Swift and Hohenberg showed that a potential V( u)can bedefined, such that the different stationary patterns correspond to local potential minima, cf.Fig (3) The dynamic of the system is first-order in time∂u/∂t = − δV/δu, where δ/δu is the

functional derivative If this energy flow stops, the velocity field in the fluid dissipates andthe nonequilibrium patterns disappear

A careful study of this system provides important insights into the behavior of nonequilibriumsystems Here, we are interested in systems for which nonequilibrium states are characterized

by non-zero values of dynamic variables A particularly simple model is discussed in Ref.(Taniguchi and Cohen 2008): a Brownian particle immersed in a viscous fluid moves at aconstant velocity under the the influence of an electric force The authors refer to it as,

a Brownian particle immersed in a fluid “NESS model of class A.” This model was usedearlier by this author to illustrate nonequilibrium stationary states (Zurcher 2008) We note,however, that this model is not appropriate to discuss important topics in nonequilibriumthermodynamics, such as pattern formation in driven-diffusive systems

Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description 7

5 Nonequilibrium stationary states: brownian particle model

Our system is a particle with mass M and the “state” of the system is characterized by the velocity v The kinetic energy plays the role of Gibbs free energy:

K(v) = M

2v

The particle at rest v = 0 corresponds to the equilibrium state, while v= const in a

nonequilibrium state We assume that the particle has an electric charge q, so that an external force is applied by an electric field Fext=qE Under the influence of this electric force, the (kinetic) energy of the particle would grow without bounds, K( t ) → ∞ for

t →∞ The coupling to a ‘thermostat’ prevents this growth of energy Here, we use avelocity-dependent force to describe the interaction with a thermostat In the terminology

of Ref (Gallavotti & Cohen 2004), our model describes a mechanical thermostat.

Dissipation is described by velocity-dependent forces, f=f(v) For a particle immersed in a

fluid, the force is linear in the velocity f l=6πaκv for viscous flow, while turbulent flow leads

to quadratic dependence f t=C0πρa2v2/2 for turbulent flow (Landau & Lifshitz 1959b) Here,

κ is the dynamic viscosity, ρ is the density of the fluid, and a is the radius of the spherical object.

These two mechanisms of dissipation are generally present at the same time; the Reynoldsnumber determines which mechanism is dominant It is defined as the ratio of inertial andviscous forces, i.e., Re=f t / f l=ρav/κ Laminar flow applies to slowly moving objects, i.e.,

small Reynolds numbers (Re<1), while turbulent flow dominates at high speeds, i.e., largeReynolds numbers (Re>105)

In the stationary state, the velocity is constant so that the net force on the particle vanishes,

Fnet=qE − f=0 We find for laminar flow,

In either case, we have Fnet > 0 for v < vsand Fnet < 0 for v > vs; we conclude that the steady

state is dynamically stable These are, of course, elementary results discussed in introductory texts, where the nonequilibrium stationary state vsis referred to as “terminal speed.”

In general, the “forces” acting on a complex system are not known, so that the timeevolution cannot be derived from a (partial-) differential equation We will show how adiscrete version of the equation can be derived from energy fluxes To this end, we recallthat in classical mechanics, velocity-dependent forces enter via the appropriate Rayleigh’sdissipation function (Goldstein 1980) We deviate from the usual definition and defineFas

the negative value of the dissipation function such that f=∂ F/∂v, and Fis associated withentropy production in the fluid If the Lagrangian is not an explicit function of time, the total

energy of the system E decreases, dE/dt = −F For laminar flow, we have

and for turbulent flow

Ft=C0π

3 ρa2v3, (turbulent flow) (14)

183Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description

The Reynolds number can be expressed as a ratio of the dissipation functions, Re∼ Ft/F l.SinceF l > Ft for v →0 andFt > F l for v →∞, we conclude that the dominant mechanismfor dissipation in the fluid maximizes the production of entropy.

The loss of energy through dissipation must be balanced by energy input in highly organized

form, i.e., work for a Brownian particle We write dW=qEdx for the work done by the electric field, if the object moves the distance dx parallel to the electric field Since v=dx/dt, the

energy input per unit time follows,

cf Ref (Zurcher 2008) This is equivalent to Newton’s second law for the object In Fig (5),

we plotF(black) andW (in blue) as a function of the velocity v The two curves intersect at

vs, so thatF = W , and the kinetic energy of the particle is constant dK/dt=0 We conclude

that vscorresponds to the nonequilibrium stationary state of the system, cf Fig (4)

The energy input exceeds the dissipated energy,F > W, for 0< v0< vs so that the excess

W − F drives the system towards the stationary state v0→ vs For v0> vs, the dissipatedenergy is higher than the input,W > F, so that the excess damping drives the system towards

vs That is, the nonequilibrium stationary state is stable,

While a mechanical thermostat allows for a description of the system’s time evolution in terms

of forces, this is not possible for an open system in contact with an arbitrary thermostat.Indeed, a discrete version of the dynamics can be found from the energy fluxesW andF

We assume that the particle moves at the initial velocity 0< v0< vs, so thatF0> W0 Wekeep the energy input fixed, and increase the velocity until the dissipated energy matchesthe input W1= F0 at the new velocity v1> v0 This first iteration step is indicated by ahorizontal arrow in Fig (5) The energy input is now at the higher valueW1> W0, indicated

by the vertical arrow By construction, the inequalityW1> F1holds, so that the procedure

can be repeated to find the the second iteration, v2, cf Fig (5) A similar scheme applies for

vs< v0<∞ We find the sequence{ v i } i for i=0, 1, 2, with v i+1< v iso that limi→∞ v i=vs.

Thus, for both v0< vsand v0> vs, the initial state converges to the stationary state,

Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description 9

We generalize this result to the case when the dissipation function is concave,

andW as a function of velocity v, cf Fig (7).

The two curves intersect at the velocity vs which characterizes the stationary state of thesystem In this case, the dissipated energy exceeds the energy input for 0< v0< vs, so that

the excess dissipation drives the system towards the equilibrium state v=0 For v0> vs, theenergy input is not balanced by the dissipated energyW > F It follows that the excess input

W − Fdrives the state of the system away from the nonequilibrium stationary state

We follow the same procedure as above, and assume that the initial velocity is (slightly) lessthan the stationary value, 0< v0< vsso thatF0> W0 We keep the energy input fixed, anddecrease the velocity untilF1= W0 at the velocity v1< v0 The iteration v0→ v1is indicated

by a horizontal arrow We now haveF1> W1, so that the steps can be repeated, cf Fig (6)

In the case v0> vs, we haveW0> F0so that the damping is not sufficient to act as a sink forthe energy input into the system Thus, the kinetic energy of the Brownian particle and the

velocity increases, v1> v0 This step is indicated by a horizontal arrow SinceW1> W0, wefindW1> F1, and the step can be repeated to find v2> v1 The corresponding phase portrait

is shown in Fig (7)

We conclude that the nonequilibrium stationary state vs is unstable when the the dissipation function is concave For v0 < vs, the initial state relaxes the towards the equilibrium state ofthe system,

6 Discussion

A Brownian particle moving in a potential well can be used toexplain some aspects

of equilibrium statistical physics We used this model to explain certain aspects ofnonequilibrium thermodynamics A nonequilibrium stationary state corresponds to the

185Nonequilibrium Thermodynamics for Living Systems: Brownian Particle Description