Gibbs canonical distribution

Choosing = as usuat then we get Gibbs canonical distribution for the occupation probabilities... 

If a system with a discrete set of its possible states s has a total energy H(s) in one of them, then the probability of the system being in the state. s is expressed by the Gibbs canonical distribution... 

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. 

Gibbs Before the interaction Ki and Kn will each be represented by a canonically distributed ensemble of gas models with moduli and n respectively. In order to represent their contact we consider the ensemble to consist of every individual of ensemble I (in its instantaneous state of motion) interacting with every individual of ensemble II. We obtain in this way a (2r,ATi- -2r,i/Vii)-dimensional T-space and in it a density distribution... 

Gibbs If one affects all the members of an initially canonical ensemble in a manner corresponding to the reversible change of the state of a gas,198 then it is permissible to assume (cf. XVIII) that the ensemble will always pass through canonical distributions only. Under these assumptions we can represent the average value over the ensemble of the 5Q defined above by199... 

Statistical theories of thermodynamics yield many correct and practical results. For example, they yield the canonical and grand canonical distributions for petit and grand systems, respectively these distributions, which were proposed by Gibbs, have been shown by innumerable comparisons with experiments to describe accurately the properties of quasistable states. Again, they predict the equality of temperatures of systems in mutual stable equilibrium, the Maxwell relations, and the Gibbs equation. 

If we select a set of points at random from the canonical distribution, then initiate trajectories of Hamiltonian dynamics from each of these points, the points will remain Gibbs-distributed over time. If the paths themselves are ergodic on the surfaces of constant energy H = E then the collection of paths may provide a usable sampling of the canonical distribution. Such a sampling technique relies on having the means of choosing initial points from the canonical distribution as this is the... 

All macroscopic observables are obtainable from the distribution of microscopic states (henceforth, microstates) of elements that obey mechanics. Strictly speaking, this is the most basic assumption of statistical thermodynamics. However, to elucidate macroscopic phenomena, it is not necessary to know the true distribution of microstates of the system. Boltzmann introduced the concept of orthodic ensembles that are compatible with thermodynamics. In practice, an orthodic ensemble is established hy demonstrating that it is compatible with the laws of mechanics and with the laws of thermodynamics. Gibbs demonstrated that the canonical distribution fimction... 

The Gibbs entropy of the canonical distribution given by Eq. (50) can be written as... 

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... 

The relationships thus established by Gibbs can be derived directly from the defining equations of these average values. They are based on the choice of the weighting function exp (4 —25)/ that is used. We should note for the discussion of question (B) that the weighting function, i.e., the canonical density distribution, depends (1) on 0 explicitly (2) through E on the ri, r, (3) through f on 0 and on the n, rs, . Let us present these results to the extent necessary for further discussion. 

Remarks on (XVI) 1. Gibbs explicitly emphasizes that he tries to prove only something about the lim (0 for < = + >, but does not want to assert that (ti.160 2. It is decisive for the later applications how low this limit turns out to be. If we admit statement (XV ) and combine it with theorem (XIHa) applied to instead of corresponding canonical p-distribution. 3. Remark (1), if applied to (XV), is identical with the question, How much time will elapse before (0 noticeably attains its limiting value ... 

Thus, in a statistically equilibrium system containing a large number of independent submacroscopic subsystems these satisfy the canonical Gibbs distribution. In this case, the following general equations describe the system ... 

Considering the system in the contact with the thermal bath (thermal reservoir) the same assumption about neglecting system-bath interaction leads to the existence of canonical (Gibbs) distribution for the probabilities to find the system in the state with energy E ... 

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