As science continues to probe systems at ever smaller length scales, the differences between the physics in the thermodynamic limit and those on the nanoscale must be explored in a rigorous manner[1]. Some of the oldest and most cherished principles of thermodynamics and statistical mechanics familiar to most scientist and engineers are only applicable in the thermodynamic limit. Consequently, care must be taken when applying macroscopic concepts to small systems, systems in which finite size effects become an important issue. For example, molecular simulation provides a set of methods that may be employed in a reasonably straightforward manner to study the behavior of nanoscale systems, in which the proper interpretation of the results of such simulations is clearly important and should be consistent with the statistical mechanics of small systems. Here, we discuss the influence of finite size effects on the principle of equipartition that is appropriate for those constraints typically imposed during molecular dynamics (MD) simulations.

The principle of equipartition, or the equipartition theorem, as it has been historically interpreted for an ergodic system, implies that each classical degree of freedom equally shares the kinetic energy of the system. Or, in other words, a single temperature can be assigned to each degree of freedom. Recently, Shirts et al.,[2] showed that the equipartition theorem breaks down for small multicomponent systems of hard spheres in the microcanonical molecular dynamics ensemble, i.e., the average kinetic energies of each species are no longer the same. The so-called molecular dynamics ensembles, which describe MD simulations with periodic boundary conditions applied, introduce two additional constraints on the system: 1) the total linear momentum is a constant of motion and 2) the center of mass of the system is a driven variable[3-5]. While the analysis of Shirts et al., is correct, their conclusion that the different components of a binary mixture must have two different temperatures is not physically consistent with results that follow from the proper form of the microcanonical ensemble.

We present theoretical and simulation work in the microcanonical, canonical (isothermal), and isothermal-isobaric molecular dynamics ensembles that provide a more transparent look into the apparent breakdown of the equipartition theorem. We begin by showing in the constant temperature molecular dynamics ensemble that the average kinetic energy of an individual particle is equal to a modified prefactor (related to the ratio of the mass of the particle to the total mass of the system) multiplied by the temperature of the bath. In other words, while the total kinetic energy of the system is no longer shared equally among the degrees of freedom, the temperature of each degree of freedom is nevertheless the same. We then consider a similar analysis for the microcanonical ensemble (which has been argued to be the natural ensemble to study small systems[1]). We obtain similar results to the constant temperature ensemble, although now the average kinetic energy of an individual particle is equal to the same modified prefactor (again related to the ratio of the mass of the particle to the total mass of the system) multiplied by the average total kinetic energy (and not the temperature). We show that each degree of freedom is still described by the same temperature and argue that the form of equipartition for the the microcanonical ensemble must be related to that ensemble's natural parameter, i.e., the total energy. Furthermore, we reconsider the so-called generalized equipartition theorem as introduced by Tolman[6], demonstrating that in all ensembles this more general form of equipartition is in fact not violated. So in the end, arguing for or against the apparent breakdown of equipartition becomes a philosophical discussion, since either viewpoint, when correctly interpreted, lead to the same physical results for small systems. Finally, we conclude with a discussion of the determination of pressure in small systems, specifically within the isothermal-isobaric molecular dynamics ensemble.

[1] D. H. E. Gross and J. F. Kenney, J. Chem. Phys., 122, 224111 (2005).

[2] R. B. Shirts, S. R. Burt, and A. M. Johnson, J. Chem. Phys., 125, 164102 (2006).

[3] T. Cagin and J. R. Ray, Phys Rev. A, 37, 247 (1988).

[4] T. Cagin and J. R. Ray, Phys Rev. A, 37, 4510 (1988).

[5] J. R. Ray and H. Zhang, Phys Rev. A, 59, 4781 (1999).

[6] R. C. Tolman, Phys Rev., 11, 261 (1918).