Canonical distribution, equilibrium phase

Hoover W G 1985 Canonical dynamics equilibrium phase-space distributions Phys. Rev. A 31 1695-7... 

Hoover W G 1985. Canonical Dynamics Equilibrium Phase-space Distributions. Physical Revic A31 1695-1697. 

W.G. Hoover, Canonical dynamics Equilibrium phase-space distributions,... 

Hoover WG (1985) Canonical dynamics equilibrium phase-space distributions. Phys Rev A 31(3) 1695—1697... 

W. G. Hoover, Phys. Rev. A, 31,1695 (1985). Canonical Dynamics Equilibrium Phase Space Distributions. 

S. Nose (1984) A unified formulation of the constant temperature molecular dynamics methods. J. Ghem. Phys, 81, pp. 511-519 W.G. Hoover (1985) Canonical dynamics Equilibrium phase-space distributions , Phys. Rev. A, 31, pp. 1695-1697... 

Hoover, W.G. Canonical dynamics equilibrium phase space distribution. Phys. Rev. 1985, A3I, 1695 1697. 

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... 

For the purposes of the present treatment, we wish to rewrite this trajectory average as an average over the initial, equilibrium distribution. If the system evolves according to deterministic (e.g., Hamiltonian) dynamics, each trajectory is uniquely determined by its initial point, and (8.46) can be written without modification as an average over the canonical phase space distribution. 

Remark. We assumed that Y(t) is a Markov process. Usually, however, one is interested in materials in which a memory effect is present, because that provides more information about the microscopic magnetic moments and their interaction. In that case the above results are still formally correct, but the following qualification must be borne in mind. It is still true that p y0) is the distribution of Y at the time t0, at which the small field B is switched off. However, it is no longer true that this p(y0) uniquely specifies a subensemble and thereby the future of Y(t). It is now essential to know that the system has aged in the presence of B + AB, so that its density in phase space is canonical, not only with respect to Y, but also with respect to all other quantities that determine the future. Hence the formulas cannot be applied to time-dependent fields B(t) unless the variation is so slow that the system is able to maintain at all times the equilibrium distribution corresponding to the instantaneous B(t). 

The quantum analogs of the phase space distribution function and the Lionville equation discussed in Section 1.2.2 are the density operator and the quantum Lionville equation discussed in Chapter 10. Here we mention for future reference the particularly simple results obtained for equilibrium systems of identical noninteracting particles. If the particles are distinguishable, for example, atoms attached to their lattice sites, then the canonical partitions function is, for a system of N particles... 

How do we calculate the probability of a fluctuation about an equilibrium state Consider a system characterized by a classical Hamiltonian H r, p ) where p and denote the momenta and positions of all particles. The phase space probability distribution isf (r, p ) = Q exp(—/i22(r, p )), where Q is the canonical partition function. 

In accordance to Gugenheim approximation the surface layer represents a zone with a certain thickness, within which the interfacial forces are not in equilibrium. Such regions appear when at least two phases interact in the following combinations liquid-gas solid-gas solid-liquid and liquid-liquid. Based on statistical physics, for the interfacial layers was defined a state function (3.224) and a probability of distribution ( 3.225 ), into a wider canonized ensamble ... 

Figure 4.5 Illustration of phase point trajectories for an equilibrium distribution in the canonical ensemble. The trajectories sweep out the entire phase space randomly. Each trajectory corresponds to a particular total energy. The probability of finding a particular phase point follows the gaussian pattern shown. Note that for a microcanonical ensemble (not shown) the probability surface is uniform or flat and each phase point trajectory is at the same fixed total energy (cf Fig. 4.1 and Fig. 4.2).

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