[Cond-Mat.Stat-Mech] 1 Oct 2001

Home , Detailed balance

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Stochastic Macromolecular Mechanics. Following Lebowitz and Spohn [8,6], we ﬁrst introduce the instantaneous heat dissipation Wt: 1 dW ≡ F (X ) ◦ dX = F (X ) · dX + dX ∇F dX (3) t t t t t 2 t t where ◦ and · denote integrations in the Stratonovich and Ito sense, respectively [9]. According to this deﬁnition, the heat dissipation, dWt, is equal to the work done by the system: the product of force F and displacement dX. This is the law of energy conservation. Using the expression (1) it is easy to show that the mean rate of the heat dissipation (HDR, hd) [8,6] d hd = E [Wt]= F (x) · J(x,t)dx (4) dt Z where

1 − J(x,t)= − A∇P (x,t) + Ξ 1F (x)P (x,t) (5) 2 is the probability flux in (2): ∂P (x,t)/∂t = −∇ · J. In mechanical terms, the mean heat dissipation rate is the product of the molecular force and the flux. The dynamic of Wt is itself Brownian-motion like with hd as its mean rate [6]. Its fluctuations can be characterized by a “heat diffusion coefficient”:

2 (dWt) −1 E =2kBT F (x)Ξ F (x)P (x,t)dx, (6) dt Z

2 which has a dimension of [energy ]/[time]. The statistical properties of Wt have been extensively explored in connection to the Gallavotti-Cohen symmetry and ﬂuctuation theorem [10,8,6]. The Onsager’s thermodynamic force is diﬀerent from the mechanical force F (x) [11]. In terms of (2), we introduce a second thermodynamic quantity, Π(x,t), the thermodynamic force [12] 1 Π(x,t) ≡ F (x) − ΞA∇ ln P (x,t), (7) 2 and a third thermodynamic quantity: the entropy S according to the well-known formula:

S = −kB P (x,t) ln P (x,t)dx (8) Z where P (x,t) is the solution to (2). In terms of (7) and (8) we have the increase of the entropy at constant temperature T

T S˙ = kB T (ln P + 1)∇ · J dx Z

= − (kBT ∇ ln P − F ) · J dx − F · J dx Z Z

= Π · Jdx − hd (9) Z = ep − hd. in which we identify, following Onsager, Π · Jdx with the entropy production rate (EPR, ep) [13]. It can also be rewritten as R T 1 −1 1 −1 ep = A∇ ln P (x,t) − Ξ F (x) Ξ A∇ ln P (x,t) − Ξ F (x) P (x,t)dx. (10) Z 2 2 which is always non-negative. This is the second law of thermodynamics. (9) is valid for all isothermal Brownian dynamical systems, with or without detailed balance, in stationary state and in transit process. It encompasses both

2 the first and second laws of thermodynamics. In fact, it makes the second law quantitative by providing a rate for entropy increase. In a time independent stationary state, the S˙ = 0 in (9), and the entropy production ep is balanced by the heat dissipation hd. This is the general case for an isothermal NESS. Eq. 10 also indicates that ep equals zero if and only if F = (ΞA/2)∇ ln P = ∇ ln P/kBT . That is the force field F has to be conservative with an internal potential energy: F = −∇U. For system with the potential, also known as detailed balance [14], the stationary solution to (2) is P = Z−1e−U/kB T where the normalization factor Z is the partition function in Gibbsian equilibrium statistical mechanics (isothermal canonical ensemble). It can be mathematically shown that ep = 0 is a sufficient and necessary condition for the stationary stochastic process Xt defined by (1) being time reversible [15]. Therefore, time reversibility, detailed balance, and zero entropy production are equivalent with an equilibrium. [16] For systems satisfying the potential condition F = −∇U, the thermodynamic force Π also has a potential: Π = −∇Ψ where Ψ(x) = U(x)+ kB T ln P . We note that the expectation of Ψt ≡ Ψ(Xt):

E [Ψt]= E [U(x)+ kBT ln P (x)] = P (x,t)U(x)dx − TS (11) Z which in fact is the Helmholtz free energy [17]! The ﬁrst term in (11) is the mean internal energy. More importantly, it is easy to show that

E Ψ˙ t = hd − T S˙ = −ep ≤ 0. (12) h i

In an isothermal system, the Helmholtz free energy decreases and reaches its minimum at the equilibrium: −kBT ln Z. This is precisely the statement of second law of thermodynamics for an isothermal system. Finally, with the potential condition, we have 1 − dU(X )= −∇U(X ) · dX − dX · ∇∇U(X ) · dX t t t 2 t t t 1 = F (X ) · dX + dX · ∇F (X ) · dX . (13) t t 2 t t t

Comparing (13) with (3), we see that the heat dissipation dWt = −dU(Xt), the internal energy ﬂuctuation. Hence, Wt = −U(Xt) is stationary and its expectation and variance are the internal energy and heat capacity (Cv) of a single macromolecule at thermal equilibrium.

The Matching Entropy and Energy Fluctuations. We now focus on systems with detailed balance. The above thermodynamic formalism suggests the fluctuating Ut ≡ U(Xt) as a mesoscopic, nonequilibrium generalization of internal energy of a macromolecule in an isothermal aqueous solution. Then its expectation E[Ut]= U(x)P (x)dx which equals to the standard internal energy in thermodynamics. Similarly in the same spirit, weR can introduce fluctuating entropy Υt ≡ −kB ln P (Xt), which can be viewed as the mesoscopic, nonequilibrium generalization of entropy. It is important to point out that this definition is consistent with the Boltzmann’s microscopic entropy based on the volume of the phase space. In our case P (Xt) is the probabilistic measure of the phase space. The expectation E[Υt] is the Gibbs entropy in Eq. 8. We now show that the mesoscopic generalizations immediately lead to an interesting thermodynamic result in equilibrium. We note that while Ut and Υt are fluctuating in equilibrium, their difference,

Ut − T Υt = U(Xt)+ kB T ln P (Xt)= −kBT ln Z (14) the equilibrium free energy, however, is not ﬂuctuating. We can further compute the ﬂuctuations in the mesoscopic entropy and internal energy:

2 2 2 T E (∆Υt) = E (∆Ut) 2 − − = U 2(x)e U(x)/kB T dx − U(x)e U(x)/kB T dx Z Z ∂ = k T 2 E[U ]= k T 2C . (15) B ∂T t B v Furthermore,

2 T E [∆Υt∆Ut]= E (∆Ut) . (16) 3 Therefore, the ﬂuctuations of the mesoscopic internal energy Ut and entropy Υt are perfectly correlated. They compensate in the dynamical ﬂuctuationsi of a macromolecule.

Entropy Fluctuation and Its Mesoscopic Interpretations. The relation between entropy fluctuation and heat capacity in Eq. 15 was known to L.D. Landau who also advocated the concept of entropy fluctuation [18]. It has been a difficulty concept to many who consider entropy to be a functional of the distribution [19]. Here we offer a more plausible interpretation of the concept based on our mesoscopic view from the previous section. Let’s consider N identical, independent macromolecules in the aqueous solution, (X1, X2, ..., XN ), each with its own stochastic dynamic equation (1). The concentration of the number of molecules in conformation x can then be defined as

N Cx,t = δ(x − Xj ). (17) Xj=1

Note that since the X’s are stochastic, the concentration Cx,t fluctuates. However, the classic statistical mechanics is only concerned with the most probable Cx,t since the relative fluctuation in Cx,t is insignificant in the thermodynamic limit when N → ∞. Nevertheless, the Cx,t fluctuates. The macroscopic entropy is defined as a functional of the density function Cx,t. Therefore, it fluctuates with the Cx,t. We now show that this fluctuation is indeed the mesoscopic fluctuation introduced in the formalism for single macromolecules. We consider a general thermodynamic quantity

q = Q(x)Cx,tdx. (18) Z

If one had neglected the ﬂuctuation in Cx,t, a ﬂuctuation in q would be inconceivable. Noting Eq. 17, the expectation of q is readily computed:

N E[q]= Q(x)dx E [δ(x − Xj )] Z Xj=1

= Q(x)dx δ(x − y )P (y , y , ..., y )dy ...dy Z Z j 1 2 N 1 N Xj=1

= N Q(x)P (x)dx = NE [Q(Xt)] Z where the joint probability for X1, X2, ...., XN , P (x1, x2, ...xN ) = P (x1)P (x2)...P (xN ) since the macromolecules are assumed to be independent in the solution. Similarly, the variance in q:

N V AR[q]= V AR Q(x)δ(x − Xj )dx Z Xj=1

N = V AR [Q(Xj )] Xj=1

= NV AR [Q(Xt)] = NV AR [Qt] in which Qt ≡ Q(Xt) [20]. Therefore, the fluctuations in q due to fluctuating in the distribution function Cx,t, is exactly the mesoscopic fluctuation we have introduced in the previous section for single macromolecules. The mesoscopic view, however, clearly indicates a stochastic dynamic origin of these fluctuations.

Entropy and Heat. It is a classic result of Clausius that entropy change equals to heat dissipation in isothermal quasi-static processes. Our result suggests that both concepts can be generalized dynamically to isothermal equilibrium, and their ﬂuctuations are indeed equal, as we have shown. In our formalism, the energy conservation is instantaneous, hence dWt = −dUt. The free energy, on the other hand, has to be constant over the entire conformational space in an equilibrium, Ψ(x) ≡ −kBT ln Z, while Υ(x) and U(x) are not. This is a demonstration

4 for the concept of local equilibrium, which is essential in the general theory on nonequilibrium thermodynamics [21]. Therefore, T dΥt = dUt = −dWt; and dUt/dΥt = T , all instantaneously.

The Rate of ep. The ep deﬁned in (10) is instantaneous and time-dependent in a nonequilibrium transient. Thus, one can further compute its time derivative (dep/dt), the change in entropy production rate [7]. With some algebra we have [22]

dep 2 −1 T −1 = −2kBT (∇ · J) P (x)dx − Π Ξ Π (∇ · J) dx. (19) dt Z Z Near a NESS, ∇ · J is small and the second term has the leading order. However, with detailed balance and near an equilibrium, Π is also small; hence the ﬁrst term, which is negative, becomes the leading term. Therefore, near an equilibrium, the ep monotonically approaches to zero. This is the Glansdorﬀ-Prigogine’s principle of minimal entropy production rate [21]. Near a NESS, however, the second term dominates, and (19) is not necessarily negative.

Discussions and Summary. While our thermodynamic formalism is strictly for isothermal processes, it can be used to compare a system under different temperature. In a thermal equilibrium, as we have shown, the fluctuation in internal energy Ut ≡ U(Xt) is directly related to the heat capacity: Cv = dE[Ut]/dT . Such relations, if any, in nonequilibrium steady-state have not been explored. Furthermore, one can easily introduce an external force into the Eq. 1 to represent the pressure, and thus we expect a thermodynamic formalism for isobaric systems can be developed in parallel. It is also noted that there is a difference between the mesoscopic energy Ut ≡ U(Xt) and entropy Υt ≡ −kB ln P (Xt). While the former can be computed along a stochastic trajectory, the latter can not until the probability distribution function P (x,t) is known. This difference reflects the fundamental difference between the two physical quantities, energy and entropy. The former is local while the latter is non-local due to circular balance: one can not know the value of entropy of a state until knowing how likely it occurrs in comparison with othe states. It is possible to formally express the P (x,t) in temrs of a path integral [23]. Maes also suggested a space-time approach to the problem [24]. In a thermodynamic equilibrium, however, P (x) can be determined locally up to a normalization factor due to detailed balance. Hence, dΥt = T dUt. In summary,

ΞdXt = F (Xt)dt +ΓdBt, (conformational dynamics)

dWt = F (Xt) ◦ dXt, (heat dissipation)

Υt = −kB ln P (Xt,t), (entropy)

Πt = F (Xt)+ T ∇Υ(Xt,t) (thermodynamic force) is a complete set of equations which provides the stochastic dynamics of a macromolecule, its heat dissipation, its entropy (Boltzmann), and its thermodynamic driving force (Onsager). With this set of equations, one can compute hd = (d/dt)E[Wt] and ep (≥ 0) from entropy balance T (d/dt)E[Υt] = ep − hd. If a system is detail balanced, i.e., F = −∇U. Then Wt = Ut ≡ U(Xt), and Π = −∇Ψ(Xt) where Ψ(Xt) ≡ Ψt = Ut − T Υt is free energy: (d/dt)E[Ψt] =-ep ≤ 0 and reaches its minimum −kBT ln Z at equilibrium. In the equilibrium, the probability distribution for Xt −1 −U(x)/kB T 2 2 2 is Z e , Ut − T Υt = −kBT ln Z, and E[(∆Υt) ] = (1/T )E[(∆Ut) ] = kBCv, the heat capacity.

ACKNOWLEDGMENTS

I thank Marty Cooksey, Bernard Deconinck, Shoudan Liang, Christian Maes, and Kyung Kim for many helpful discussions.

[1] F. J¨ulicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997); H. Qian, J. Math. Chem. 27, 219 (2000). [2] H. Qian, LANL E-print, physics/0007017 (2000). [3] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986).

5 [4] Models with nonconservative forces have been developed for molecular motors and their stationary solutions represent NESS in which the chemical energy in the form of ATP hydrolysis is converted to mechanical work. This class of models opens the vista for studying mesoscopic dynamics and thermodynamics at NESS. See [1,2] and H. Qian, Phys. Rev. Lett. 81 3063 (1998). [5] H. Qian and J.J. Hopfield, J. Chem. Phys. 105, 9292 (1996); H. Qian, J. Chem. Phys. 109, 10015 (1998); R.M. Levy and E. Gallicchio, Ann. Rev. Phys. Chem. 49, 531 (1998). [6] H. Qian, Phys Rev. E. 64, 022101 (2001). [7] J. Honerkamp, Statistical Physics: An Advanced Approach with Applications (Springer, Berlin, 1998). [8] J.L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333 (1999). [9] B. Øksendal, Stochastic Differential Equation (4th Ed., Springer, New York, 1995). [10] G. Gallavotti, Phys. Rev. Lett. 77, 4334 (1996); G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995). [11] L. Onsager, Phys. Rev. 37, 405 (1931). [12] The Π corresponds to the affinity in the master equation formalism Aij = kB T ln(pj qji/piqij ) where {qij } is the transition rate and pi the state i probability. Similarly, the “mechanical work” F (Xt) ◦ dXt corresponds to kB T ln(qji/qij ). See [13,6] for more details. R [13] J. Schnakenberg, Rev. Mod. Phys. 48, 571 (1976). [14] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, (Springer-Verlag, New York, 1984). [15] M-P. Qian, M. Qian, and G.L. Gong, Comtemp. Math. 118, 255-261 (1991); H. Qian, Proc. R. Soc. Lond. A. 457, 1645 (2001). [16] In the literature, the notion of detailed balance has two closely related, but different meanings. For a stationary Markov process, it means the transition flux from i to j equals exactly the transition flux from j to i; hence the net flux is zero. This notion can be mathematically shown to be equivalent to time-reversibility. Stationarity does not imply time-reversibility. Detailed balance is also used as a property of a Markovian system which, when reaching its stationary state, is time reversible with zero flux. The second notion can be mathematically shown to be equivalent to the potential condition. By the laws of thermodynamics, an equilibrium has to have detailed balance; and a system without energy input has to be detail balanced. [17] It can be shown that the free energy Ψ is equivalent to the relative entropy. The latter is well-known as a Lyapunov function for the stochastic dynamics (2) [14,13]. For a physical interpretation of the relative entropy, see H. Qian, Phys. Rev. E. 63, 042103 (2001). [18] L.D. Landau and E.M. Lifshitz, Statistical Physics (3rd Ed., Part 1, Pergamon Press, Oxford, 1980). [19] A. Cooper, Progr. Biophys. Mol. Biol. 44, 181 (1984). [20] In terms of the probability theory, Xk are independent, identially distributed random variables. Hence Q(Xk) are independent, identically distributed. Therefore, the expectation and variance of the sum of Q(Xk) is the sum of the expectation E[Q(Xk)] and variance V AR[Q(Xk)]. [21] P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations. (Wiley Intersciences, New York, 1971). [22] The first term in (19) is associated with 2 (∂Π/∂t) · Jdx and the second term with Π · (∂J/∂t)dx − (∂Π/∂t) · Jdx. See J.-l. Luo, C. Van den Broeck, and G. Nicolis,R Z. Phys. B. 56, 165 (1984). R R [23] H. Haken, Zeit. Phys. B 24, 321 (1976). [24] C. Maes, Statistical mechanics of entropy production: Gibbsian hypothesis and local fluctuations. preprint (2001).