Boltzmann distribution, canonical ensemble

Consider the generalized distribution Pq(r ) to be generated in the Gibbs-Boltzmann canonical ensemble (9 = 1) by an effective potential W,(r /3) which is defined... 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

The distribution modulus, 9 of the canonical ensemble thus possesses the property of an empirical temperature 9 = kT. The proportionality constant, by analogy with ideal gas kinetic theory, is the same as Boltzmann s constant. 

The interpretation of the above expressions is rather remarkable. The centroid constraints in the Boltzmann operator, which appear in the definition of the QDO from Eqs. (19) and (20), cause the canonical ensemble to become non-stationary. Equally important is the fact that the non-stationary QDO, when traced with the operator 9. (or P) as in Eq. (37), defines a dynamically evolving centroid trajectory. The average over the initial conditions of such trajectories according to the centroid distribution [ cf Eq. (36) ] recovers the stationary canorrical average of the operator (or ). However, centroid trajectories for individual sets ofirritial conditions are in fact dynamical objects and, as will be shown in the next section, contain important information on the dynamics of the spontaneous fluctuations in the canonical ensemble. 

According to (XII) it is first of all plausible that in general the average (Eq. 62) over the canonical ensemble will be very nearly identical with the average value taken over a microcanonical or even ergodic ensemble with E=E0> In fact, in that case also Eq. (57), for example, coincides with a relationship derived by Boltzmann (1871) for ergodic ensembles.182 Furthermore, the micro-canonical ensemble is very nearly equivalent to an ensemble that is distributed (cf. Section 12c) with constant density over the shell in T-space belonging to... 

If we now supplement Gibbs s discussion with the investigations of Boltzmann as presented in Section 13(1), we come to the following conclusion In a canonically distributed ensemble of gas models the overwhelming majority of the individual members are in a state described by the Maxwell-Boltzmann distribution given in Eq. (46) with the parameters n, , rm, and with the energy E—E. 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

As mentioned earlier, the limiting probabilities occur only in the ratio pjpm and the value of the denominator, Qc, is therefore not required. Hence, in the canonical ensemble, the acceptance of a trial move depends only on the Boltzmann factor of the energy difference, AU, between the states m and n. If the system looses energy, then the trial move is always accepted if the move goes uphill, that is, if the system gains energy, then we have to play a game of chance on exp(—fi AU) [6]. In practical terms, we have to calculate a random number, uniformly distributed in the interval [0,1]. If < exp(—p AU), then we accept the move, and reject otherwise. 

The summation includes all possible stationary states with energy , that are occupied by the molecule. The equilibrium probability Pi that the molecule populates each of these states is given by a Boltzmann distribution for the canonical ensemble. In other words. 

In this section, we focus on a relation between entropy and the probability distribution function P,. If P, obeys Boltzmann statistics for the canonical ensemble, then one arrives at a correspondence between entropy S and the partition function Z. Equation (28-26) is interpreted within the context of the first law of thermodynamics in differential form ... 

When the Boltzmann distribution was derived in Section 5.2, we assumed a constant number of particles (N) and a given volume (V). We also assumed that the energy was constant. In the canonical ensanble, we imagine that the subsystems (members of the canonical ensemble) may exchange energy but not particles with each other. The ensemble that we are supposed to use is the one where the assumptions agree with the experimental conditions. Since the total energy is constant in the microcanonical ensemble, the definition of such a simple concept as temperature has to be done indirectly. 

The equilibrium distribution in the canonical ensemble, a closed system containing just N molecules, is implied by the proportionality of fiN.Ni Boltzmann factor of the Hamiltonian Xu,... 

Mb is interpreted as the mass of the heat bath . For appropriate choices of Mb, the kinetic energy of the particles does indeed follow the Maxwell-Boltzmann distribution, and other variables follow the canonical distribution, as it should be for the AfVT ensemble. Note, however, that for some conditions the dynamic correlations of observables clearly must be disturbed somewhat, due to the additional terms in the equation of motion [(38) and (39)] in comparison with (35). The same problem (that the dynamics is disturbed) occurs for the Langevin thermostat, where one adds both a friction term and a random noise term (coupled by a fluctuation-dissipation relation) [75, 78] ... 

The canonical ensemble is one in which the system is in contact with a heat bath. In the dynamical picture, one envisions that the system of interest is part of a much larger system that is isolated such that the total energy of the composite system is fixed. The energy of the subsystem (the system of interest) can fluctuate. The probability of a given microstate in this subsystem can be calculated by using energy conservation (of the total system) and f2i2(F = i + 2) = where 1 and 2 refer to the subsystem and the bath. The result is the famous Boltzmann distribution Py = where P is the temperature of the bath... 

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