# Thermodynamics and Fluctuations Far From Equilibrium

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. One-variable Systems

#### 2.1. Linear One-variable Systems

_{1}and k

_{2}for the forward and reverse reaction of the first reaction ($A\iff X$) and k

_{3}and k

_{4}the corresponding rates for the second reaction. In this sequence A is the reactant, X the intermediate, and B the product. Let the reactions occur in the schematic apparatus of Figure 1 at constant temperature.

_{X}is given by:

**Figure 1.**Two piston model. The reaction compartment (II) is separated from a reservoir of species A (I) by a membrane permeable only to A and from a reservoir of species B (III) by a membrane permeable only to B. The pressures of A and B are held fixed by constant external forces on the pistons. Catalysts C and C’ are required for the reactions to occur at appreciable rates and are contained only in region II.

_{II}is a volume shown in Figure 1. This function has many important properties. At the stationary state of this system, $\mathrm{\varphi}$ is zero. If we start at the stationary state and increase p

_{X}then dp

_{X}≥ 0 and the integrand is larger than zero; hence $\mathrm{\varphi}$ is positive. Similarly, if we start at the stationary state and decrease p

_{X}, then dp

_{X}and the integrand are both negative and $\mathrm{\varphi}$ is positive. Hence $\mathrm{\varphi}$ is an extremum at the stable stationary state, a minimum.

#### 2.2. Nonlinear One-variable Systems

_{1}and k

_{2}for the forward and reverse reaction in (7), and k

_{3}and k

_{4}in (8). For this system there exists the possibility of multiple stationary states for given constraints of the pressures p

_{A}and p

_{B}. The kinetic equation for p

_{X}is:

_{X}and hence may have three stationary states (RHS of (9) equals zero), as shown in Figure 2.

**Figure 2.**Stationary states of the Schlögl model with fixed reactants and products pressures. Plot of the pressure of the intermediate p

_{X}vs. the pump parameter (p

_{A}/p

_{B}). The branches of stable stationary states are labeled α and γ and the branch of unstable stationary states is labeled β. The marginal stability points are at F

_{1}and F

_{3}and the system has two stable stationary states between these limits. The equistability point of the two stable stationary states is at F

_{2}.

_{A}, p

_{B}, T, V

_{I}, V

_{II}, V

_{III}, and equilibrium constants for the A⇔X and B⇔X reactions. Then the two model systems are instantaneously thermodynamically equivalent. If furthermore ${t}_{X}^{+}$ and ${t}_{X}^{-}$ have the same values in the two systems at each point in time, the two systems are instantaneously kinetically indistinguishable. If we define ${p}_{X}^{\ast}$ as the pressure of X in the instantaneously indistinguishable linear system at stationary state, we may write from (4) and (5):

_{X}compared to the work of moving the system from the stationary state of the instantaneous indistinguishable linear system to p

_{X}. If A and B in (6) are chosen such that the ratio of their pressures equals the equilibrium constant, then ${\mathrm{\varphi}}^{*}$ = ΔG and ${p}_{X}^{\ast}={p}^{s}$.

#### 2.3. Dissipation

_{X}:

_{X}, regardless of the reaction mechanism; the equality holds only at the stationary state.

_{X}is such an extremum and the integral:

#### 2.4. Connection of the Thermodynamic Theory with Stochastic Theory

_{i}related to the rate coefficients k

_{i}by ${k}_{i}={V}^{{m}_{i}-1}({c}_{i}/{n}_{t}!)$ for $1\le i\le 4$, where m

_{i}is the molecularity of the i

^{th}step and n

_{i}the molecularity in X.

_{1}molecules in V at time t

_{1}given that there are X

_{0}molecules at t

_{0}. This function is the solution of the master equation (26) for the initial condition $P(X,t={t}_{0})=\delta (X-{X}_{0})$. The probability density can be factored into two terms [3]:

_{0}to X

_{1}and independent of the time interval (t

_{1}–t

_{0}). To the same approximation with which we obtained (29) we can reduce the first term to:

#### 2.5. Relative Stability of Multiple Stationary Stable States

**Figure 3.**Plot of the integral in (30), marked ${\mathrm{\varphi}}_{s}$ vs. X for the Schlögl model with parameters: c

_{1}= 3.10

^{−10}s

^{−1}; c

_{2}= 1.10

^{−7}s

^{−1}; c

^{3}= 0.33 s

^{−1}; c

_{4}= 1.5.10

^{−4}s

^{−1}; and A = B. For curve (a) B = 9.8.10

^{6}; for curve (b) B = 1.01.10

^{6}; curve (c) B = 1.04.10

^{6}. Curve (b) lies close to the equistability of the stable stationary states 1 and 3; 2 marks the unstable stationary state.

_{1}to X

_{2}to that of the reverse transition:

^{ss}that number in the stationary state, ${P}_{S}(X)=\left[{({X}^{SS})}^{X}/X!\right]\mathrm{exp}(-{X}^{SS})$. Our results are consistent with (36), as can be seen from the use of (15) and (30), a change of variables to particle numbers X, and the use of Stirling’s approximation:

^{ss}). The formulation given in this section has the advantage of the physical interpretation in terms of species-specific thermodynamic driving forces and in terms of Lyapunov functions; further our formulation is generalizable to autocatalytic systems and many variable systems.

## 3. Thermodynamic State Function for Multivariable Systems

_{X}is the probability distribution of finding

**X**particles (molecules) in a given volume, and W(

**X,r**) is the transition probability due to reaction from

**X**to

**X+r**particles. Now we do a Taylor expansion of the term $W\left(X-r,r\right)P\left(X-r,t\right)$ around

**X**:

_{X}(

**X,**t) is set to zero, of the form:

_{n}will be shown to be the classical action of a fluctuational trajectory accessible from the n

^{th}stable stationary state. We substitute (45) into the stationary part of (43) and obtain:

_{n}(

**X**) with the Hamiltonian function ( not operator):

_{n}that satisfies the Hamiltonian-Jacoby equation (46) with coordinate

**x**, momentum $p=\partial {S}_{n}(x)/\partial x$, and Hamiltonian equal to zero (stationary condition). The Hamiltonian equations of motion for the system are:

^{th}stable stationary state

**x**

_{n}

^{s}with

**p**= 0 at t = −$\infty $ and ending at

**x**at t = 0.

#### 3.1. Linear Multi-variable Systems

_{s}, Y

_{s}are stable stationary state concentrations. We see that the integrand is an exact differential and hence the action is independent of the path of integration in concentration space. The action is a state function [8,9].

_{s}, the system in the stationary state. Hence we have:

**x,r**) are all positive and the square bracket is larger than zero except for

**p**= 0, that is at the stable stationary state. Therefore we have:

**r·p**the square bracket in (60) is negative unless

**p**= 0. And therefore:

#### 3.2. Nonlinear Multi-variable Systems

**p**and d

**x/**dt obtained from solutions of Hamilton’s equations. We now choose our reference by using the equations:

^{0},Y

^{0}) in the absence of certain crossings of fluctuational trajectories in the (X,Y) space, called ‘caustics’, see [10]. There may be more than one fluctuational trajectory which starts at

**p**= 0 at a stable stationary state and passes through a given (X,Y). These trajectories will have different values of

**p**and the one with the lowest value of

**p**will determine the action in the thermodynamic limit, the contributions from other trajectories vanishing in that limit.

^{0},Y

^{0}) replaces the starred reference state of section 2, see Equation (13). The important point is that the action and the excess work in (64) are state functions for single and multi-variable systems. Both X

^{0}and Y

^{0}are functions of X and Y in general, but the integrand in (64) is an exact differential, because

**p**is the gradient of the action, (51). For the starred reference state the excess work is a state function only for single variable systems.

^{∗}equals $\varphi $

^{0}(64). In summary, we define the state function $\varphi $

^{0}with the use of (64):

^{0}is a thermodynamic state function. It is a potential for the stationary probability distribution of the master equation, and is a Lyapunov function in the domain of each stable stationary state. See also [10,11,12,13,14,15,16,17]. It is directly related to the excess work necessary to remove a system from a stable stationary state, and the work obtainable from a system in its return to such a state. It is an extremum at stationary states; a minimum (zero) at stable stationary states, a maximum at unstable stationary states (59). For a fluctuational trajectory $\varphi $

^{0}increases away from the stable stationary state (59); for a deterministic trajectory towards a stable stationary state it decreases, (60). The first derivative of $\varphi $

^{0}is larger than zero at each stable stationary state, smaller than zero at each unstable stationary state. The function ${\varphi}^{0}$ provides necessary and sufficient criteria for the existence and stability of stationary states. $\varphi $

^{0}serves to determine relative stability of multi-variable homogeneous systems in exactly the same way as shown in (33) for single variable systems. Comparison with experiments on relative stability requires consideration of space-dependent (inhomogeneous) systems and that subject is discussed in the next section.

**Figure 4.**From [7]. S1 and S3 are stable stationary states (stable foci); S2 denotes an unstable stationary state. The solid line from S2 to S3 indicates the deterministic trajectory. The other solid line through S2 is the deterministic separatrix, that is the line that separates deterministic trajectories, on one side going towards S2 and on the other side going towards S3. The dotted lines are fluctuational trajectories: one from S3 to S2 and the others proceeding from S2 in two different directions. The fluctuational trajectory need not differ so much from the reverse of the deterministic trajectory.

^{0}, Y

^{0}requires solution of the master equation for a particular reaction mechanism. This in general demands numerical solutions, which can be lengthy. However, if we do not need to obtain a thermodynamic function, this is not necessary and the deterministic approach of section 2 may be used instead.

^{0}can be determined from macroscopic electrochemical measurements, as well as other measurements, see section 6.3.

## 4. Thermodynamic and Stochastic Theory of Reaction-diffusion Systems

#### 4.1. Systems with One Intermediate

^{+}, t

^{−}are kinetic fluxes. If the system is inhomogeneous, the concentrations are functions of time and spatial coordinates. Let us consider a one-dimensional system with spatial coordinate z and discretize the space into many boxes labeled with …i−1, i, i+1, as in Figure 5. The change in the number of particles X

_{i}is due to reaction and diffusion into and out of box i and can be written:

**Figure 5.**Schematic apparatus for reaction-diffusion system in one spatial dimension. The boxes 1 to N are separated from a constant-pressure reservoir of A by a membrane permeable only to A, and similarly for the reservoir of B [18].

^{2}d, where l is the length of a box. Then in the continuous limit we have:

_{i}, is:

_{i}, is an independent variable, a reaction-diffusion system is isomorphic to a multivariable homogeneous reaction system, which can be linearized around a stable stationary state. At that state we have ${\frac{d\varphi}{d{X}_{i}}|}_{\mathrm{det}}^{S}=0$, where ‘det’ means that the deterministic path is the path of integration. The time derivative of $\varphi $ satisfies $\frac{d\varphi}{dt}\le 0$. $\varphi $ is a minimum at stable stationary states and a Lyapunov function in their vicinity.

#### 4.2. Systems with Two Intermediates

_{i}, Y

_{i}) we can uniquely map the non-linear system to a thermodynamically and kinetically equivalent linear system. In this case the total excess work is:

#### 4.3. Relative Stability of Stable Stationary States

_{6}, which is the rate coefficient corresponding to the transformation from B to Y, and the ratio of diffusivities of X and Y, δ = D

_{y}/D

_{x}.

**Figure 6.**Plots of concentration profiles of X and Y vs. distance z during the front propagation in the Selkov model. The solid line is the initial concentration profile; the dotted lines are the concentration profiles with uniform time spacing.

**Figure 7.**Selkov model: the solid line is a plot of zero velocity of the interface between phase 1 and 3. Above the solid line the interface moves to the right, below it to the left.

**Figure 8.**Comparison of predictions of equistability from the thermodynamic theory (b, c, d) with the numerical solution (a) repeated from Figure 7. The theoretical solutions correspond to different lengths of the interface region L: (b) 6L, (c) 2L, (d) L. The theoretical curves approach the numerical calculations as the length of the interface region, and the number of boxes, are increased.

#### 4.4. Stability and Relative Stability of Reaction-diffusion Systems Related to Fluctuations

^{0}. This presentation is based on the results in [17,20]. For a system with two intermediates (x,y) we can write:

^{0},y

^{0}) refer to a reference state given by ${p}_{x}=\mathrm{ln}(x/{x}^{0}),\text{\hspace{0.17em}}{p}_{y}=\mathrm{ln}(y/{y}^{0}),$ which hold for an equivalent linear system. The displacements on the RHS of (86) are along the most probable fluctuational path. The momentum p is the gradient of the action, and therefore ds and d$\varphi $

^{0}are exact differentials. Let us divide the interphase region into N boxes; then the state function of the total excess work is the sum of that in each box:

^{0}. If the stationary state 1 (SS1) is slightly more stable than the stationary state 3 (SS3) then the deterministic motion of the front is a translation to the right. Since $\varphi $

^{0}is a Lyapunov function Δ$\varphi $

^{0}for this process must be negative. Similarly for the opposite case, 3 slightly more stable than 1, Δ$\varphi $

^{0}is also negative. Hence at equistability the limiting value of Δ$\varphi $

^{0}for a translation along the position z must be zero.

**Figure 9.**Concentration (X) versus position (z) for the Selkov model. The initial concentration profile is shown by the solid line; the space with negative z is filled initially with stationary state 1, the space with positive z is filled initially with stationary state 3. The dotted line denotes the interphase region.

#### 4.5. An Experiment on Relative Stability of Multiple Stationary States

^{−3}s

^{−1}for the flow rate coefficient at zero velocity of propagation. This experimental result was compared with a calculation based on a further simplification of the NFT mechanism to a two-variable system [24]. Numerical solution of the corresponding deterministic reaction-diffusion equations yields the value of 12.2 × 10

^{−3}s

^{−1}, while the thermodynamic theory of section 4.3 predicts a value of 12.45 × 10

^{−3}s

^{−1}. In view of the limited precision of the experiments and the use of a very simple model of the reaction mechanism, the agreement of the experiment with the theory and the calculations is satisfactory.

**Figure 10.**Plots of the measured dependency of the velocity of front propagation of one stationary state into the other on the flow rate, k

_{f}. The eight different symbols correspond to eight experiments. Two of the plots are shifted from their original positions for the purpose of better display: circles by −0.6 cm/min in V; diamonds by −0.7 cm/min in V. Lines are fitted to each set of points for purpose of extrapolation to zero velocity of front propagation. The precision of the points at the largest velocities is insufficient to permit extrapolation.

## 5. Thermodynamic and Stochastic Theory of Transport Processes

#### 5.1. Linear Thermal Conduction

_{1}and the other at T

_{3}, with T

_{1}> T

_{3}. Volume 2 is between the two thermal reservoirs and is of small width so that its temperature T is uniform within it. The flow of heat occurs with conservation of energy and no work done.

_{1}, dQ, dQ

_{3}. The integral of d$\varphi $ is:

_{1}and k

_{3}are proportional to thermal conduction coefficients for the interface of the system with the heat reservoirs 1 and 3, respectively. The temperature of the stationary state is ${T}_{S}=\frac{{k}_{1}{T}_{1}+{k}_{3}{T}_{3}}{{k}_{1}+{k}_{3}}$, and hence we may write $dT/dt=({k}_{1}+{k}_{3})({T}_{S}-T)$. If we divide by dt in (90) and use $\frac{dQ}{dt}={C}_{V}dT$, we obtain for the time derivative of $\varphi $:

_{S}.

_{S}we arrive at the expected quadratic form:

_{S}= T

_{equ}. For an ideal gas we have E = C

_{V}T and the fluctuations are in the Gaussian form in energy:

#### 5.2. An Experiment on Optical Bistability

**Figure 12.**Two possible stable stationary states of an optically bistable interference filter. The region of irradiation is the length L; the ambient, lower, and upper temperatures are T

_{0}, T

_{1}, and T

_{3}.

_{3}toward T

_{0}. Calculations are shown on the left side of Figure 13 [26]; they are confirmed by the experimental measurements on the right side. The reaction-diffusion theory is amended for a problem in thermal conduction [25], yielding a calculation of the irradiation power at equistability can of 1.375 mW. This shows a very good agreement with the experimental measurement, which was 1.395 ± 0.015 mW.

**Figure 13.**Calculated (left) and measured (right) plots of the decay of temperature profiles on stoppage of irradiation of the upper stationary state (

**a**) and the lower stationary state (

**b**).

#### 5.3. Coupled Transport Processes: An Approach to Thermodynamics and Fluctuations in Hydrodynamics

#### 5.3.1. Lorenz Equations and an Interesting Experiment

_{c}, then the zero solution is unique and stable, and it corresponds to the motionless conductive state of the fluid. At the bifurcation point r = 1 this solution becomes unstable, and a new solution becomes stable corresponding to convective modes. These solutions can be used to construct an excess work function, just as we did for single transport properties.

#### 5.3.2. Rayleigh Scattering in a Fluid in a Temperature Gradient

**Figure 14.**The product of the total rate of dissipation times temperature (solid line) in J/s and the time derivative of excess work (dashed line) vs. time in the following processes for the Lorenz model: (a) Gravity is initially set in the direction along which the temperature decreases, and the system is at a stable motionless conductive stationary state; at t = 0, invert the direction of gravity; the motionless conductive state becomes unstable and the system approaches the convective stationary state. (b) The reverse process. The temperature difference is $\left|\Delta T\right|=4K$ in both cases.

## 6. Electrochemical Experiments in Systems Far from Equilibrium

#### 6.1. Measurement of Electrochemical Potentials in Non-equilibrium Stationary States

_{ss}in the stationary state is obtained from the absorption measurement. The local equilibrium emf, the third column in Table 1 is calculated from the ratio Ce(III)/Ce(IV) and the Nernst equation. The measured emf in the fourth column of the table is that measured by the Pt electrode. The difference is small, about 1% of the emf at the largest inflow concentration of Ce(III)

_{0}and decreases for smaller inflow concentrations.

**Table 1.**Results from the Minimal Bromate Experiment with Various Concentrations of Ce(III) in the combined feedstreams into the Reactor, [Ce(III)]

_{0}. From [50].

[Ce(III)]_{0} (M) | [Ce(III)]_{ss} | IE emf (mV) | M emf (mV) |
---|---|---|---|

1.500 × 10^{−3} | 1.393 × 10^{−3} | 1167.0 | 1180.2 |

1.397 × 10^{−3} | 1.361 × 10^{−3} | 1176.0 | 1183.0 |

1.297 × 10^{−3} | 1.277 × 10^{−3} | 1187.5 | 1189.7 |

8.333 × 10^{−4} | 8.700 × 10^{−4} | 1223.7 | 1223.8 |

#### 6.2. Kinetic and Thermodynamic Information Derived from Electrochemical Measurements

^{III}, 0.0100 M BrO

_{2}

^{−}, and 1.00 × 10

^{−6}M Br

^{−}, and each reservoir contains also 0.72 M H

_{2}SO

_{4}. To run the reaction at equilibrium the three solutions are mixed and allowed to react for a day prior to being pumped into the CSTR. First we measure the Ce(III)/Ce(IV) potential with zero imposed current from an external current source; then we impose various currents and displace the equilibrium mixture in the CSTR from equilibrium. A non-equilibrium stationary state is achieved by flowing the reacting solutions into the CSTR at given flow rates, that is given residence times in the reactor. For a residence time of 175 seconds, Figure 15 shows the measured voltages plotted against the imposed current, as well as the Ce(IV) concentration, and the product of the measured voltage minus the stationary state voltage multiplied by the imposed current. The left hand part of Figure 15 shows results for an equilibrium mixture, while the right hand part shows displacements from a non-equilibrium stationary state.

_{ss}) × I in the equilibrium case is nearly symmetric, but that for the non-equilibrium cases it is not.

**Figure 15.**Plot of voltage V, the power input (V − V

_{ss}) × I, and the Ce(IV) concentration versus the imposed current I. V

_{ss}is the measured voltage at a non-equilibrium stationary state at zero imposed current. The plot on the left corresponds to an equilibrium stationary state; the residence time in the CSTR is 200 seconds. The plot on the right hand corresponds to a non-equilibrium stationary state at zero imposed current and displacements from that state with imposed currents; the residence time is 175 s. The arrows indicate transitions to other stationary states. From [51].

#### 6.3. Theory of Determination of Thermodynamic and Stochastic Potentials from Macroscopic Measurements

_{c}, is (see Section 3):

_{1}, B

_{1}, taken to be neutral, and the species A

_{2}, B

_{2}taken to be negatively charged. The differential d$\varphi $

_{c}is exact and any path of integration suffices to obtain $\varphi $. The exponential of the integral in (106) is a formal representation of the eikonal approximation for the stationary solution of the master equation of the chemical system.

_{2}, is ${\mu}_{{A}_{2}}+{E}_{{A}_{2}}NF,$ where E

_{A2}is the potential for a given imposed current, N is the number of equivalents in the half-cell reaction for A

_{2}, and F is the Faraday constant. We postulate that we may write d$\varphi $

_{E}for the combined chemical and electrochemical system in a parallel way:

_{E}= 0 and therefore we have for d$\varphi $

_{c}the result:

_{A2(s)}of A and E

_{B2(s)}of B in the stationary state of the system of the chemical system. The first term in each square bracket depends on concentrations only and thus is the Nernstian contribution to the measured electrochemical potential. The second term in each square bracket depends on the kinetics of the chemical system and thus is the non-Nernstian contribution the electrochemical potential. Measurements of this potential, say with ion-specific electrodes, yield the slopes $\frac{\partial {\varphi}_{c}}{\partial {n}_{{A}_{2}}}$ and $\frac{\partial {\varphi}_{c}}{\partial {n}_{{B}_{2}}}$; thus with measurements of the macroscopic concentrations of A

_{1}, A

_{2}, B

_{1}, and B

_{2}at a sufficient number of displacements from the stationary state of the chemical system we can determine the stochastic potential of that system from macroscopic measurements. To obtain these results no direct use of any master equation has been made and no model of the reaction mechanism was necessary.

#### 6.3.1. Determination of the Stochastic Potential in Chemical Systems with Imposed Fluxes

_{1}, J = k’A

_{1}

^{’}, into the reaction chamber with Q

_{+}and Q

_{−}held constant and thereby move the chemical system to a different non-equilibrium stationary state with different concentrations of the reacting species A

_{1}, B

_{1}, A

_{2}, B

_{2}. This procedure allows the sampling of different combinations of the reacting species by means of the imposition of different fluxes of these reactants. These combinations represent different non‑stationary states in the absence of imposed fluxes, but with the imposed fluxes they are stationary states and hence measurements may be made without constraints of time. If we would attempt to measure concentrations in non-stationary states then the measurement technique would have to be fast compared to the time scale of change of the concentrations due to chemical reactions.

**, as well. For multivariable systems this approach is more difficult; the determination of the stochastic potential requires sufficient measurements to determine rate coefficients and then the numerical solution of the stationary form of the master equation. Details of this procedure are described in Appendix A of [52].**

^{−}#### 6.3.2. Suggestions for Experimental Tests of the Master Equation

**Figure 16.**Typical hysteresis loop for a one-variable system with a cubic kinetic equation: plot of concentration c vs. influx coefficient. Solid lines, stable stationary states (nodes); broken line, unstable stationary state. For a discussion of lines A and B and numbers, see text.

## 7. Dissipation in Irreversible Processes

#### 7.1. Exact Solution for Thermal Conduction

_{1}and T

_{2}; we assume T

_{1}> T

_{2}. We take the conduction of heat to be given by Newton’s equations:

#### 7.2. Invalidity of the Principles of Minimum and Maximum Entropy Production

## 8. Efficiency of Oscillatory Reactions

**Figure 17.**Plot of ΔG and the rate of the HRP reaction vs. time. See text for further explanation. From [58].

**Figure 18.**Plot of the phase difference between the Gibbs free energy change and the rate of the HRP reaction in units of 2π. From [58].

## 9. Fluctuation-Dissipation Relations

^{+}(x) − t

^{−}(x)

^{+}(x) + t

^{−}(x)]/ 2.

_{x}− μ

^{s}

_{x}) for the linear case or (μ

_{x}− μ

^{*}

_{x}) for the non-linear case, the driving force for the reaction towards a stationary state. Thus we have a flux–driving force relation. Secondly, the formulation is symmetric with respect to t

^{+}(x) and t

^{−}(x), which is not the case with other formulations. Third, the state function $\varphi $ determines the probability distribution of fluctuations in x from its value at the stationary state, see (30). Further, as we shall show shortly, the term D(x) is a measure of the strength of the fluctuations of the total number of reaction events (total in the forward and reverse reaction). We emphasize that (127) is non-linear and holds for arbitrarily large fluctuations. Expressions for different regimes of affinity and dissipation can be easily obtained [64,65,66].

## Acknowledgements

## Appendix: Elementary Thermodynamics and Kinetics

_{j}for each of the j reactions:

_{i}denotes number of moles of species i, and the affinities A

_{j}[67]:

^{th}reaction the kinetics can be written as $d{\xi}_{j}/dt={t}_{j}^{+}-{t}_{j}^{-}(1\le j\le J)$, where ${t}_{j}^{+},\text{\hspace{0.17em}}{t}_{j}^{-}$ are the reaction fluxes for this reaction step in the forward and reverse direction, respectively. Hence the affinities may be rewritten as ${A}_{j}=RT\mathrm{ln}({t}_{j}^{+}/{t}_{j}^{-})$, which is easily obtained for any elementary reaction by writing out the ${t}_{j}^{+}/{t}_{j}^{-}$ in terms of concentrations and the introduction of chemical potentials. The time rate of change of the Gibbs free energy is:

## References and Notes

- Ross, J. Thermodynamics and Fluctuations Far from Equilibrium; Springer: Berlin, Germany, 2008. [Google Scholar]
- Van KampenN.G.Stochastic Processes in Physics and Chemistry; North-Holland: Amsterdam, The Netherlands, 1981.
- Ross, J.; Hunt, K.L.C.; Hunt, P.M. Thermodynamics far from equilibrium: Reactions with multiple stationary states. J. Chem. Phys.
**1988**, 88, 2719–2729. [Google Scholar] [CrossRef] - Schlögl, F. On thermodynamics near a steady state. Z. Phys.
**1971**, 248, 446–458. [Google Scholar] [CrossRef] - Evans, D.J.; Searles, D.J.; Williams, S.R. On the fluctuation theorem for the dissipation function and its connection with response theory. J. Chem. Phys.
**2008**, 128, 014504. [Google Scholar] [CrossRef] [PubMed] - Glansdorff, P.; Nicolis, G.; Prigogine, I. The thermodynamic stability theory of non-equilibrium states. Proc. Natl. Acad. Sci. U. S. A.
**1974**, 71, 197–199. [Google Scholar] [CrossRef] [PubMed] - Peng, B.; Hunt, K.L.C.; Hunt, P.M.; Suárez, A.; Ross, J. Thermodynamic and stochastic theory of nonequilibrium systems: Fluctuation probabilities and excess work. J. Chem. Phys.
**1995**, 102, 4548–4562. [Google Scholar] [CrossRef] - Oppenheim, I.; Shuler, K.E.; Weiss, G.H. Stochastic Processes in Chemical Physics: The Master Equation; MIT: Cambridge, MA, USA, 1977. [Google Scholar]
- Gardiner, C.W. Handbook of Stochastic Methods; Springer: New York, NY, USA, 1990. [Google Scholar]
- Nicolis, G.; Babloyantz, A. Fluctuations in open systems. J. Chem. Phys.
**1969**, 51, 2632–2637. [Google Scholar] [CrossRef] - Selkov, E.E. Self-oscillations in glycolysis. J. Biochem.
**1968**, 4, 79–86. [Google Scholar] - Graham, R.; Tél, T. Nonequilibrium potential for coexisting attractors. Phys. Rev. A
**1986**, 33, 1322–1337. [Google Scholar] - Maier, R.S.; Stein, D.L. Transition-rate theory for nongradient drift fields. Phys. Rev. Lett.
**1992**, 69, 3691–3695. [Google Scholar] [CrossRef] [PubMed] - Crandall, M.G; Evans, L.C.; Lions, P.L. Some Properties of Viscosity Solutions of Hamilton‑Jacobi Equations. Trans. AMS
**1984**, 282, 487–502. [Google Scholar] - Jauslin, H.R. Nondifferentiable potentials for nonequilibrium steady states. Physica A
**1987**, 144, 179–191. [Google Scholar] [CrossRef] - Jauslin, H.R. Melnikov's criterion for nondifferentiable weak-noise potentials. J. Stat. Phys.
**1986**, 42, 573–585. [Google Scholar] [CrossRef] - Freidlin, M.I.; Wentzell, A.D. Random Perturbations of Mechanical Systems; Springer: Berlin, Germany, 1984. [Google Scholar]
- Chu, X.; Ross, J.; Hunt, P.M.; Hunt, K.L.C. Thermodynamic and stochastic theory of reaction‑diffusion systems with multiple stationary status. J. Chem. Phys.
**1993**, 99, 3444–3454. [Google Scholar] [CrossRef] - Hansen, N.F.; Ross, J. Lyapunov functions and relative stability in reaction-diffusion systems with multiple stationary states. J. Phys. Chem.
**1996**, 100, 8040–8043. [Google Scholar] [CrossRef] - Hansen, N.F.; Ross, J. Relative stability of multiple stationary states related to fluctuations. J. Phys. Chem.
**1998**, 102, 7123–7126. [Google Scholar] [CrossRef] - Glauber, R.J. Lectures in Theoretical Physics; Interscience Publishing: New York, NY, USA, 1959. [Google Scholar]
- Noyes, R.M.; Field, R.J.; Thompson, R.C. Mechanism of reaction of bromine (V) with weak one‑electron reducing agents. J. Am. Chem. Soc.
**1971**, 93, 7315–7316. [Google Scholar] [CrossRef] - Foerster, Y.-X.; Zhang, J.; Ross, J. Experiments on relative stability in the bistable multivariable bromate-ferroin reaction. J. Phys. Chem.
**1993**, 97, 4708–4713. [Google Scholar] [CrossRef] - Bar-Eli, K; Geiseler, W. Mixing and relative stabilities of pumped stationary states. J. Phys. Chem.
**1981**, 85, 3461–3468. [Google Scholar] - Wolff, A.N.; Hjelmfelt, A.; Ross, J.; Hunt, P.M. Tests of thermodynamic theory of relative stability in one-variable systems. J. Chem. Phys.
**1993**, 99, 3455–3460. [Google Scholar] [CrossRef] - Harding, R.H.; Ross, J. Experimental measurement of the relative stability of two stationary states in optically bistable ZnSe interference filters. J. Chem. Phys.
**1990**, 92, 1936–1946. [Google Scholar] [CrossRef] - Lamb, H. Hydrodynamics, 6th ed.; Dover Publications: New York, NY, USA, 1945. [Google Scholar]
- Saltzman, B. Finite amplitude free convection as an initial value problem-I. J. Atmos. Sci.
**1962**, 19, 329–341. [Google Scholar] [CrossRef] - Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] - Getling, A.V. Rayleigh-Bénard Convection. Structure and Dynamics; World Scientific Pub. Co.: London, UK, 1997. [Google Scholar]
- Glansdorff, P.; Prigogine, I. Thermodynamic Theory of Structure, Stability, and Fluctuations; Wiley: New York, NY, USA, 1971. [Google Scholar]
- Keizer, J. Statistical Thermodynamics of Nonequilibrium Processes; Springer-Verlag: New York, NY, USA, 1987. [Google Scholar]
- Attard, P. The second entropy: A general theory for non-equilibrium thermodynamics and statistical mechanics. Annu. Rep. Prog. Chem. Sect. C Phys. Chem.
**2009**, 105, 63–173. [Google Scholar] [CrossRef] - Ronis, D.; Procaccia, I. Nonlinear resonant coupling between shear and heat fluctuations in fluids far from equilibrium. Phys. Rev. A
**1982**, 26, 1812–1815. [Google Scholar] [CrossRef] - Fox, R.F. Gaussian stochastic processes in physics. Phys. Rep.
**1978**, 48, 179–283. [Google Scholar] [CrossRef] - Velarde, M.G.; Chu, X.-L.; Ross, J. Toward a thermodynamic theory of hydrodynamics: The Lorenz equations. Phys. Fluids.
**1994**, 6, 550–563. [Google Scholar] [CrossRef] - Zamora, M.; Rey de Luna, A. Energy of a system formed by a convective fluid and its container. J. Fluid Mech.
**1986**, 167, 427–437. [Google Scholar] [CrossRef] - Law, B.M.; Sengers, J.V. Fluctuations in fluids out of thermal equilibrium. J. Stat. Phys.
**1989**, 57, 531–547. [Google Scholar] [CrossRef] - Law, B.M.; Segrè, P.N.; Gammon, R.W.; Sengers, J.V. Light-scattering measurements of entropy and viscous fluctuations in a liquid far from thermal equilibrium. Phys. Rev. A
**1990**, 41, 816–824. [Google Scholar] [CrossRef] [PubMed] - Segrè, P.N.; Gammon, R.W.; Sengers, J.V.; Law, B.M. Rayleigh scattering in a liquid far from thermal equilibrium. Phys. Rev. A
**1992**, 45, 714–724. [Google Scholar] [CrossRef] [PubMed] - Li, W.B.; Segrè, P.N.; Gammon, R.W.; Sengers, J.V. Small-angle Rayleigh scattering from nonequilibrium fluctuations in liquids and liquid mixtures. Physica A
**1994**, 204, 399–435. [Google Scholar] [CrossRef] - Suárez, A.; Ross, J. Thermodynamic and stochastic theory of coupled transport processes: Rayleigh scattering in a fluid in a temperature gradient. J. Phys. Chem.
**1995**, 99, 14854–14863. [Google Scholar] [CrossRef] - Ross, J.; Chu, X.-L. Thermodynamic and stochastic theory for nonideal systems far from equilibrium. J. Chem. Phys.
**1993**, 98, 9765–9770. [Google Scholar] [CrossRef] - Berry, R.S.; Rice, S.A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: New York, NY, USA, 2000. [Google Scholar]
- Keizer, J.; Chang, O.-K. The nonequilibrium electromotive force. I. Measurements in a continuously stirred tank reactor. J. Chem. Phys.
**1987**, 87, 4064–4073. [Google Scholar] - Keizer, J. The nonequilibrium electromotive force. II. Theory for a continuously stirred tank reactor. J. Chem. Phys.
**1987**, 87, 4074–4087. [Google Scholar] - Keizer, J. Thermodynamics at isothermal, isobaric steady states: vapor pressure, colligative properties, and the electromotive force. J. Phys. Chem.
**1989**, 93, 6939–6943. [Google Scholar] [CrossRef] - Keizer, J. Heat, work, and the thermodynamic temperature at non‑equilibrium steady states. J. Chem. Phys.
**1985**, 82, 2751–2771. [Google Scholar] [CrossRef] - Field, R.J.; Burger, M. Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, NY, USA, 1985. [Google Scholar]
- Hjelmfelt, A.; Ross, J. Electrochemical experiments on thermodynamics at nonequilibrium steady states. J. Phys. Chem.
**1994**, 98, 9900–9902. [Google Scholar] [CrossRef] - Hjelmfelt, A.; Ross, J. Kinetic and thermodynamic information derived from electrochemical measurements on stationary states. J. Phys. Chem. B
**1998**, 102, 3441–3444. [Google Scholar] [CrossRef] - Ross, J.; Hunt, K.L.C.; Vlad, M.O. Determination of thermodynamic and stochastic potentials in non-equilibrium system from macroscopic measurements. J. Phys. Chem. A
**2002**, 106, 10951–10960. [Google Scholar] [CrossRef] - Ross, J.; Hunt, K.L.C.; Hunt, P. Thermodynamics and stochastic theory for non-equilibrium systems with multiple reactive intermediates: The concept and role of excess work. J. Chem. Phys.
**2002**, 96, 618–629. [Google Scholar] [CrossRef] - Kramer, J.; Ross, J. Stabilization of unstable states, relaxation, and critical slowing down in a bistable system. J. Chem. Phys.
**1985**, 83, 6234–6241. [Google Scholar] [CrossRef] - Prigogine, I. Moderation et transformations irreversibles des systemes ouverts. Bull. Cl. Sci., Acad. R. Belg.
**1945**, 31, 600–606. [Google Scholar] - Ross, J.; Vlad, M.O. Exact solutions for the entropy production rate of several irreversible processes. J. Phys. Chem. A
**2005**, 109, 10607–10612. [Google Scholar] [CrossRef] [PubMed] - Martyushev, L.M.; Seleznev, V.D. Maximum entropy production principle in physics, chemistry, and biology. Phys. Rep.
**2006**, 426, 1–45. [Google Scholar] - Lazar, J.G.; Ross, J. Changes in mean concentration, phase shifts, and dissipation in a forced oscillatory reaction. Science
**1990**, 247, 189–192. [Google Scholar] [CrossRef] [PubMed] - Lax, M. Fluctuations in a nonequilibrium stable state. Rev. Mod. Phys.
**1960**, 32, 25–64. [Google Scholar] [CrossRef] - Lax, M. Classical noise III: Nonlinear markoff processes. Rev. Mod. Phys.
**1966**, 38, 359–379. [Google Scholar] [CrossRef] - Lax, M. Classical noise IV: Langevin methods. Rev. Mod. Phys.
**1966**, 38, 541–566. [Google Scholar] [CrossRef] - Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill: New York, NY, USA, 1965; Chapters 1, 15. [Google Scholar]
- Chandrasekhar, S. Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys.
**1943**, 15, 1–89. [Google Scholar] [CrossRef] - Vlad, M.O.; Ross, J. Fluctuation-dissipation relations for chemical systems far from equilibrium. J. Chem. Phys.
**1994**, 100, 7268–7278. [Google Scholar] [CrossRef] - Vlad, M.O.; Ross, J. Random paths and fluctuation-dissipation dynamics for one-variable chemical systems far from equilibrium. J. Chem. Phys.
**1994**, 100, 7279–7294. [Google Scholar] [CrossRef] - Vlad, M.O.; Ross, J. Thermodynamic approach to nonequilibrium chemical fluctuations. J. Chem. Phys.
**1994**, 100, 7295–7309. [Google Scholar] [CrossRef] - Nicolis, G.; Prigogine, I. Self-organization in Nonequilibrium Systems; Wiley: New York, NY, USA, 1977. [Google Scholar]
- Tolman, R.C. The Principles of Statistical Mechanics; Oxford University Press: London, UK, 1938. [Google Scholar]

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Ross, J.; Villaverde, A.F.
Thermodynamics and Fluctuations Far From Equilibrium. *Entropy* **2010**, *12*, 2199-2243.
https://doi.org/10.3390/e12102199

**AMA Style**

Ross J, Villaverde AF.
Thermodynamics and Fluctuations Far From Equilibrium. *Entropy*. 2010; 12(10):2199-2243.
https://doi.org/10.3390/e12102199

**Chicago/Turabian Style**

Ross, John, and Alejandro Fernández Villaverde.
2010. "Thermodynamics and Fluctuations Far From Equilibrium" *Entropy* 12, no. 10: 2199-2243.
https://doi.org/10.3390/e12102199