J. Phys. Chem. 1996, 100, 9149-9151
9149
Extremum Behavior of Fluctuation Amplitudes Close to Equilibrium Andra´ s Baranyai*,† and Peter T. Cummings Department of Chemical Engineering, UniVersity of Tennessee, 419 Dougherty Engineering Building, KnoxVille, Tennessee 37996-2200, and Chemical Technology DiVision, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6268 ReceiVed: December 14, 1995X
In a previous paper (Phys. ReV. E 1995, 52, 2198) we proposed that the fluctuation amplitudes of equilibrium systems are minimal relative to nearby dissipative nonequilibrium states. This behavior was found in nonequilibrium molecular dynamics calculations at fixed density and kinetic temperature. In this paper we examined this rule at fixed internal energy (microcanonical ensemble) and also for two-dimensional systems. We find that the extremum behavior remains valid despite the change in boundary conditions and system dimensionality.
Introduction paper1
we examined the fluctuations of wellIn a previous defined nonequilibrium steady state (NESS) systems. We proposed that the equilibrium state of the system has the smallest fluctuations relative to nearby nonequilibrium states. Nonequilibrium molecular dynamics (NEMD) systems served as test cases to check the validity of this hypothesis. We found that the proposed principle is verified for these dissipative systems if the volume, V, the number of particles, N, and the temperature, T, were held fixed. In the present paper we want to examine what kind of “nearby nonequilibrium states” one can expect to be subjected to this extremum principle. In this respect the most important question is the ensemble dependence of the proposed principle. There are an infinite number of systematic ways one can select nearby nonequilibrium states of an equilibrium system. For instance, we can imagine a model liquid containing simple, classical particles interacting with spherically symmetric pairwise additive forces. Using the standard Metropolis Monte Carlo method one can sample the equilibrium configurations of the system. In this scheme the size of the Boltzmann factor, exp(-β∆Φ), where β ) 1/kT and ∆Φ is the change of the internal energy, governs the occurrence of the individual configurations.2 After some initial period, the accepted configurations provide an accurate representation of the equilibrium liquid. One can distort this process by slightly changing the Boltzmann factor. An obvious way to do this by assuming a hard core diameter for the particles and accepting random moves if there is no overlap between the particles, but rejecting them otherwise. If the hard core diameter is smaller than the effective size of the particles the Markov process will frequently attain high-energy configurations very rarely reached by the correct equilibrium scheme. The amplitude of the fluctuations calculated for this nonequilibrium set of configurations will be larger than the equilibrium value. On the other hand, if one chooses a large hard core diameter the attainable set of configurations will be constrained with smaller fluctuation amplitudes than in equilibrium. In these processes we generated nonequilibrium states by changing the weight of configurations relative to their equilibrium value. Obviously, nonequilibrium states produced † Permanent address: Eo ¨ tvo¨s University, Laboratory of Theoretical Chemistry, Budapest 112, PO BOX 32., 1518-Hungary. X Abstract published in AdVance ACS Abstracts, March 15, 1996.
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by similar stochastic recipes can have smaller or larger fluctuations based on the selection procedure and will not obey the proposed extremum rule. The simulated models (planar Couette flow and “color” flow) of the previous paper1 represented steady state dissipatiVe systems driVen by mechanical forces giVen explicitly in the equations of motion. A large body of theoretical study and computational evidence shows, however, that they are congruent with realistic, boundary-driVen systems.3 It is reasonable to assume that the same sort of fluctuation behavior can be expected in the latter cases as well. If one removes the steady driving force from these systems they relax back to equilibrium. It is also reasonable to assume that fluctuation amplitudes of these transient states are also larger than in equilibrium. A different questionsnot studied in the previous paper1sis the ensemble dependence of the proposed principle. The calculations reported in ref 1 were carried out at constant volume, V, number of particles, N, and temperature, T, the latter being identified with the kinetic or equipartition temperature of the system. Thus, the equilibrium system was a member of the canonical ensemble. It is intriguing to ask the question whether the proposed principle remains valid in the microcanonical ensemble also, when instead of the temperature the internal energy, E is fixed. The answer is far from obvious because the kinetic temperature of the system decreases with increasing external field at constant internal energy while the internal energy for constrained kinetic temperature increases with increasing external field. Thinking in terms of statistical mechanical ensembles helps us to understand the behavior of other transient states. For instance, one can prepare nonequilibrium transient states by removing external, conservative force fields like gravitation or electrostatic from the system. At t ) 0 the system is in equilibrium but under the impact of the external field. The instantaneous removal of the field causes the system to relax back to its field-free equilibrium state. It is again reasonable to assume that the fluctuation amplitudes of the transient states can be interpolated between those of the field-on and the fieldfree equilibrium states. Since the field-on states can have either larger or smaller fluctuations than the field-free states depending on the type of the external field and the boundary conditions, the transients generated by the removal of conservative external fields, in general, will not obey the proposed extremum principle. © 1996 American Chemical Society
9150 J. Phys. Chem., Vol. 100, No. 21, 1996
Baranyai and Cummings
We emphasize that the conclusions of the present study refer primarily to hydrodynamic nonequilibrium steady state systems and should not be regarded as generally applicable to nonequilibrium systems, but rather applied with considerable caution. Evidence for this is provided by Keizer,4 who provides an extensive discussion of fluctuations in nonequilibrium steady state systems. For example, Keizer reports no evidence of extremum behavior in theoretical studies of bistable chemical systems, and fluctuations in these close-to-equilibrium systems can be larger or smaller than at equilibrium depending on how the steady state is achieved. There are also electrochemical processes where fluctuation amplitudes are not minimized at equilibrium.4 Figure 1. Temperature as a function of the shear rate. (N ) 500, T ) 0.722, F ) 0.8442, WCA particles).
Results and Discussion To study the validity of the proposed extremum principle, we performed simulations of a three-dimensional system at fixed internal energy, E, volume, V, and number of particles, N. The parameters of the calculations were chosen to be close to our previous fixed kinetic temperature, T, simulations.1 The internal energy of the system was fixed by a differential feedback scheme using the well-known SLLOD algorithm.3 As in ref 1, we considered a system of 500 particles interacting via the so-called WCA potential. This spherically symmetric, pairwise additive short-range interaction is defined in terms of the interparticle separation, r, as follows:
φ(r) ) 4[r-12 - r-6] + 1 )0
r e 21/6 r g 21/6
Here and throughout the paper we use the usual reduced units of computer simulations (distances made dimensionless by dividing by the molecule diameter σ, energies made dimensionless by dividing by the characteristic interaction energy , temperature made dimensionless by multiplying by k/ where k is Boltzmann’s constant, number densities made dimensionless by multiplying by σ3, strain rates made dimensionless by multiplying by (mσ2/)1/2, and times made dimensionless by dividing by (mσ2/)1/2.2 The state point of the system is also given in reduced units. The number density, F ) 0.8442 was the same as in ref 1. The internal energy per particle was chosen to be E/N ) 1.85. We used the time step of 0.004. A fifthorder Gear algorithm integrated the equations of motion. The equilibration period was 100 000 time steps for each calculation. The length of each equilibrated run was 1 million time steps, i.e. 4000 reduced time units. The errors in our results were calculated by dividing each simulation run into 20 blocks. The error bars correspond to one standard deviation of the block averages. In the figures the error bars shown are the same for each data point because the average of the error bars is shown rather than the individual error bars for each shear rate. This simplification could be made because individual error bar values were found to be close to one another. The variance of a quantity was obtained by calculating the quantity 〈A2〉 - 〈A〉2 in each block from the individual time steps in the block and then analyzing the block averages to obtain the overall average value and the standard deviation in this quantity. In Figure 1 we show the temperature of the model fluid as a function of the shear rate. The decrease of the kinetic temperature in terms of the shear rate is considerable. If we examine the variance of the temperature in Figure 2, there is no doubt that the shear rate increases the fluctuations of this
Figure 2. Temperature fluctuation as a function of the shear rate. The system is the same as in Figure 1.
Figure 3. Shear stress fluctuation as a function of the shear rate. The system is the same as in Figure 1.
quantity. In Figure 3 the variance of the stress is shown as a function of the shear rate. For the sake of comparison the first few data points of our previous calculations1 at fixed kinetic temperature are also presented. Although the error bars are larger for the fixed energy system and the increase of the fluctuations is weaker, one can reasonably conclude that the fluctuations have a minimum at equilibrium even if the internal energy is fixed. [The chosen value of the fixed internal energy, 1.85 is very close to the internal energy of the constant temperature (T ) 0.722) system at shear rate 0.5 (1.852) so these two points overlap on the figure.] We also performed calculations for two-dimensional fluids because, in many respects, these systems behave differently from three-dimensional ones. From our present point of view the most relevant difference is the divergence of the transport coefficients given by Green-Kubo expressions in twodimensional liquids.5 This divergence is special to twodimensional systems and reflects the importance of fluctuations relative to surface effects. However, it is also known that
Extremum Behavior of Fluctuation Amplitudes
J. Phys. Chem., Vol. 100, No. 21, 1996 9151
TABLE 1: Properties of a Two-Dimensional System at Different Shear Rates, γ, and at Constant Kinetic Temperaturea γ
E/N
(δE/N)2 × 103
-Pxy
(δPxy)2 × 103
0.00 0.02 0.05 0.10 0.15 0.20 0.25 0.30
1.1362 1.1360 1.1363 1.1364 1.1367 1.1371 1.1375 1.1387
1.665 2.378 3.323 3.858 4.332 3.947 5.218 4.746
0.0000 0.0118 0.0312 0.0599 0.0899 0.1180 0.1475 0.1757
6.604 6.643 6.673 6.688 6.765 6.777 6.893 7.090
a E/N is the internal energy per particle; (δE/N)2 is the variance of the internal energy; Pxy is the shear stress; (δPxy)2 is the variance of the shear stress. (T ) 1.00, N/V ) 0.50, N ) 400) The error in the energy variance is (1.06 × 10-3. The error in the stress variance is (0.162 × 10-3.
TABLE 2: Properties of a Two-Dimensional System at Different Shear Rates, γ, and at Constant Internal Energya γ
T
(δT)2 × 104
-Pxy
(δPxy)2 × 103
0.00 0.02 0.05 0.10 0.20 0.30 0.40 0.50
1.1426 1.1426 1.1424 1.1426 1.1418 1.1404 1.1389 1.1369
4.278 4.285 4.288 4.255 4.318 4.457 4.573 4.727
0.000 0.0124 0.0323 0.0631 0.1237 0.1850 0.2434 0.2997
8.230 8.312 8.321 8.305 8.320 8.557 8.749 8.907
a T is the kinetic temperature; (δT)2, the variance of the kinetic temperature; Pxy, the shear stress; (δPxy)2, the variance of the shear stress. (E/N ) 1.30, N/V ) 0.50, N ) 400) The error in the temperature variance is (0.078 × 10-4. The error in the stress is (0.315 × 10-3.
despite the theoretically based divergences computer simulations of two-dimensional systems yield sensible data.6 Our results, summarized in Tables 1 and 2, are consistent with this observation. The results reported in Tables 1 and 2 were obtained from 1 million time step runs for each state point with a time step of 0.004. The proposed extremum behavior of the fluctuations appears to remain valid in these dilute, twodimensional systems. The minimum was more pronounced in the constant temperature case as opposed to the constant energy case, just as in the three-dimensional results. Unfortunately, we could not prove our hypothesis. To prove it one needs to go beyond the linear regime because the change in fluctuations is at least second order in the external field. To our knowledge, there is no generally accepted nonlinear response theory which could predict such a behavior. A simple rationalization for the extremum behavior of fluctuations, however, can be given by the minimum entropy production principle.7 The principle claims that in the linear regime the system opts for that state which has the smallest entropy production. This
means, at a given thermodynamic force, the system adopts the state with the smallest dissipative flux which is ensured by the minimum value of the corresponding transport coefficient. The value of the transport coefficient can be determined from the Green-Kubo relation as the integral of the autocorrelation function of the flux in question.3 The value of the autocorrelation function at t ) 0 is the fluctuation amplitude of the particular property. Assuming that the shape of the autocorrelation function remains unchanged close to equilibrium, the larger starting value will cause ultimately larger entropy production. This qualitative reverse argument emphasizes that the validity of the proposed fluctuation extremum has the same root as the well-known minimum entropy principle.7 A weakness in this rationalization is the fact that both the minimum entropy principle and the Green-Kubo relations have validity only in the linear regime where, since the fluctuations being second order in terms of the external field, they are indistinguishable from the equilibrium value. A few years ago, Evans and Baranyai proposed an extremum principle for ergodic, nonequilibrium steady states (NESS) far from equilibrium.8 This hypothesis is a microscopic and nonlinear generalization of the principle of minimum entropy production which is valid only for NESS close to equilibrium where both the local thermodynamic equilibrium and the Onsager reciprocal relations are expected to hold.7 Theoretical calculations have not verified this hypothesis for external fields of arbitrary size (the exactness of the principle does not hold beyond the linear regime),9 but even if the EB principle is only an approximation it seems a very powerful one outside but close to the linear regime. Acknowledgment. A.B. thanks the financial support of OTKA Grant No. F7218. P.T.C. gratefully acknowledges partial support of this research by the National Science Foundation through grant CTS-9101326 and by Lockhead-Martin Energy Systems at Oak Ridge National Laboratory through the Distinguished Scientist program. References and Notes (1) Baranyai, A.; Cummings, P. T. Phys. ReV. E 1995, 52, 2198. (2) Allen, M. P.; Tildesley, D. J. Computer simulation of liquids; Clarendon Press: Oxford, 1987. (3) Evans, D. J.; Morriss, G. P. Statistical mechanics of nonequilibrium liquids; Academic Press: New York, 1990. (4) Keizer, J. Statistical Thermodynamics of Nonequilibrium Processes; Springer-Verlag: Berlin, 1987. (5) Ernst, M. H.; Hauge, E. H.; van Leeuwen, J. M. J. Phys. ReV. Lett. 1970, 25, 1254. (6) Hoover, W. G.; Posch, H. A. Phys. ReV. E 1995, 51, 273. (7) de Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover: New York, 1984. (8) Evans, D. J.; Baranyai, A. Phys. ReV. Lett. 1991, 67, 2597. (9) Santos, A.; Garzo´, V.; Brey, J. J. Europhys. Lett. 1995, 29, 693.
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