Microcanonical equilibrium

Figure 12. The diffusive modes of the periodic Yukawa-potential Lorentz gas represented by their cumulative function depicted in the complex plane ReFk,hnFk) for two different nonvanishing wavenumbers k. The horizontal straight line is the curve corresponding to the vanishing wavenumber k = 0 at which the mode reduces to the invariant microcanonical equilibrium state.

In this chapter we will extend the fluctuation-dissipation theorem for general equilibrium distributions. We will consider two typical equilibrium distributions. One is the superstatistical equilibrium distribution [13]. The other is the microcanonical equilibrium distribution. 

The other distribution is the microcanonical equilibrium distribution. More than 15 years ago, Ott-Brown-Grebogi pointed out fractional scaling of deviation from ergodic adiabatic invariants in Hamiltonian chaotic systems [16, 17]. We will reconsider not only ergodic adiabatic invariants but also nonergodic adiabatic invariants, which are important in the mixed phase space. We will show results of our numerical simulation in which a nonergodic adiabatic invariant corresponding to uniform distribution is broken in the mixed phase space. 

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

We will first consider the fluctuation-dissipation theorem for the microcanonical distribution. The microcanonical equilibrium distribution is given as... 

With regard to the microcanonical equilibrium distribution and the extension of the fluctuation-dissipation theorem, we considered a nonergodic adiabatic invariant in a simple Hamiltonian chaotic system. We numerically demonstrated the breaking of the nonergodic adiabatic invariant in the mixed phase space. The variance of the nonergodic adiabatic invariant can be considered as a measure for complexity of the mixed phase space. 

Figure 9. Time evolution of average fractional energy in model OCS. Here the total energy is 20,000 cm 1 with almost all energy initially in the bend mode

This is the microcanonical equilibrium constant. The canonical counterpart is written in terms of the partition functions. 

The forward ( ab) nd reverse ( ba) constants at long times are related to the microcanonical equilibrium constant K by the well-known rela-... 

If there are several local minima of N(E,s) as a function of s, then this corresponds to several "bottlenecks" of the reactive flux. If one assumes that microcanonical equilibrium is established locally in the regions between these bottlenecks—e.g., by existence of long-lived intermediates—then one can derive a "unified" statistical... 

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

An explicit example of an equilibrium ensemble is the microcanonical ensemble, which describes closed systems with adiabatic walls. Such systems have constraints of fixed N, V and E < W< E + E. E is very small compared to E, and corresponds to the assumed very weak interaction of the isolated system with the surroundings. E has to be chosen such that it is larger than (Si )... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

In the last subsection, the microcanonical ensemble was fomuilated as an ensemble from which the equilibrium, properties of a dynamical system can be detennined by its energy alone. We used the postulate of... 

For practical calculations, the microcanonical ensemble is not as useful as other ensembles corresponding to more connnonly occurring experimental situations. Such equilibrium ensembles are considered next. 

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... 

Suppose now that we have an ensemble of N non-interacting particles in a thermally insulated enclosure of constant volume. This statement means that the number of particles, the internal energy and the volume are constant and so we are dealing with a microcanonical ensemble. Suppose that each of the particles has quantum states with energies given by i, 2,... and that, at equilibrium there are Ni particles in quantum state Su particles in quantum state 2, and so on. 

Calvo, F. Neirotti, J.P. Freeman, D.L. Doll, J.D., Phase changes in 38-atom Lennard-Jones clusters. II. A parallel tempering study of equilibrium and dynamic properties in the molecular dynamics and microcanonical ensembles, J. Chem. Phys. 2000, 112, 10350-10357... 

Therefore, if g — g(a), the ensemble represents a steady state or equilibrium distribution. The two most important steady-state distributions are known as microcanonical and canonical ensembles. 

Thus, in equilibrium the system spends on the average equal times in each of the Q. states. The calculation of the time average can therefore be replaced by averaging over the quantum statistical microcanonical ensemble. However, as in classical theory, equating time average with microcanonical ensemble average remains conjectural. 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

The microcanonical ensemble in quantum statistics describes a macroscopi-cally closed system in a state of thermodynamic equilibrium. It is assumed that the energy, number of particles and the extensive parameters are known. The Hamiltonian may be defined as... 

The adaptation is such as to permit the equilibrium microcanonical distribution for the slow coordinate X to be a solution (2.3) when k(X) = 0. The SU(X) in Eq. (2.3) is the vibrational entropy change needed to reach X from... 

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,... 

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