Equilibrium statistical mechanics phase space

The aim of this section is to give the steady-state probability distribution in phase space. This then provides a basis for nonequilibrium statistical mechanics, just as the Boltzmann distribution is the basis for equilibrium statistical mechanics. The connection with the preceding theory for nonequilibrium thermodynamics will also be given. 

First let us assume that the system has been undisturbed for so long that it is in a macrostate of thermal equilibrium. Trajectories will then pass through the bottleneck region equally often from left to right and from right to left, and the probabilities of different microstates in the bottleneck region, as in any part of phase space, will be given by the formulas of equilibrium statistical mechanics (e.g. the equilibrium microcanonical density,... 

From the earlier section on Molecular Dynamics and Equilibrium Statistical Mechanics, we hope that we have made clear that a conserved quantity is the starting point for phase space analysis and the derivation of a probability distribution function. Following the same analysis that led to the distribution functions for NVE, NVT, and NPT dynamics, the new distribution /(q, p,, I) for GSLLOD coupled to a Nose-Hoover thermostat is given by... 

The maximum information entropy procedure is the derivation of the Gibbs ensemble in equilibrium statistical mechanics, but the information entropy is not defined by a probability measure on phase space, but rather on path space. The path a information entropy is... 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

In statistical mechanics of equilibrium one assigns equal probabilities to equal volume elements of the energy shell in phase space. 510 This assignment is determined... 

The profound consequences of the microscopic formulation become manifest in nonequilibrium molecular dynamics and provide the mathematical structure to begin a theoretical analysis of nonequilibrium statistical mechanics. As discussed earlier, the equilibrium distribution function / q contains no explicit time dependence and can be generated by an underlying set of microscopic equations of motion. One can define the Gibbs entropy as the integral over the phase space of the quantity /gq In / q. Since Eq. [48] shows how functions must be integrated over phase space, the Gibbs entropy must be expressed as follows ... 

In statistical mechanics we do not measure observables directly. Instead we observe an average over all possible values. The averaging is done by means of a probability distribution function, which in classical mechanics is averaged over all of phase space. Let us compare an equilibrium ensemble with grand potential Q, and an arbitrary nearby ensemble prepared by a small perturbation, AQ. Let the equilibrium probability distribution function be f and that for the nearby... 

Under very general conditions, it follows from classical statistical mechanics that the equilibrium behavior of our fluid system is adequately described % the behavior of a Gibbskn ensemble of systems characterized by a canonical distribution (in energy) in phase space. This has two immediate consequences. First it specifies the spatial distribution of our N molecule system. The simultaneous probability that some first molecule center hes in the volume element dr whose center is at and etc., and the Nih molecule center lies in the volume element dr f whose center is at is... 

The main postulate of statistical mechanics states that, for an equilibrium system of given mass, composition, and spatial extent, all microstates with the same energy are equally probable [37,38]. This postulate, along with the ergodic hypothesis, can be justified on the basis of the mixing flow in phase space exhibited by the dynamical trajectories of real systems [28]. It means that all microstates of an isolated system, which does not exchange energy and material with its environment, should occur equally often. 

In transport calculations, as first approximation one can linearize in the external disturbances represented by / (k, r), i.e., one can perform a linear approximation in the field strengths E and B (small fields). Since the equilibrium distribution does not carry any net flow, one must calculate the true distribution at least to first order in the driving forces. From statistical mechanics one knows that phase points move in the (k, r) space like an incompressible fluid, thus the total time derivative of... 

We consider only the equilibrium case so that the distribution of these points phase space is time-independent. In quantum statistical mechanics, we had a discrete list of possible states. In classical statistical mechanics, we have coordinates and momentum components that can range continuously. We denote the probability disttibution (probability density) for the ensemble by / and define the probability that the phase point of a randomly selected system of the ensemble will lie in the 6A -dimensional volume element d tNci pi to be... 

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