Canonical distribution

The above derivation leads to the identification of the canonical ensemble density distribution. More generally, consider a system with volume V andA particles of type A, particles of type B, etc., such that N = Nj + Ag +. . ., and let the system be in themial equilibrium with a much larger heat reservoir at temperature T. Then if fis tlie system Hamiltonian, the canonical distribution is (quantum mechanically)... 

The canonical distribution corresponds to the probability density for the system to be in a specific microstate with energy E- H, from it one can also obtain the probability P( ) that the system has an energy between E and E + AE i the density of states D E) is known. This is because, classically. 

If one denotes the averages over a canonical distribution by (.. . ), then the relation A = U-T S and U = (W) leads to the statistical mechanical coimection to the themiodynamic free energy A ... 

The T-P ensemble distribution is obtained in a maimer similar to the grand canonical distribution as (quantum mechanically)... 

As an alternative to sampling the canonical distribution, it is possible to devise equations of motion for which the iiiechanicaT temperature is constrained to a constant value [84, 85, 86]. The equations of motion are... 

Given a size N lattice (thought of now as a heat-bath), consider some subsystem of size n. An interesting question is whether the energy distribution of the subsystem, Pn E), is equal to the canonical distribution of a thermodynamic system in equilibrium. That is, we are interested in comparing the actual energy distribution... 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

The angular brackets ( ) in (7.33) denote an average over an ensemble of nonequilibrium transformation processes initiated from states z distributed according to a canonical distribution. The Jarzynski identity (7.33) is valid for nonequilibrium transformations carried out at arbitrary speed. 

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

In the standard HMC method two ingredients are combined to sample states from a canonical distribution efficiently. One is molecular dynamics propagation with a large time step and the other is a Metropolis-like acceptance criterion [76] based on the change of the total energy. Typically, the best sampling of the configuration space of molecular systems is achieved with a time step of about 4 fs, which corresponds to an acceptance rate of about 70% (in comparison with 40-50% for Metropolis MC of pure molecular liquids). 

Equations (8.44) and (8.45) guarantee convergence to a canonical distribution only in the case of fixed B. Because B varies (i.e., the method uses information from momenta sampled in the past in determining the vector B), the evolution is not strictly Markovian. As a consequence, the correlations introduced can lead to the accumulation of systematic errors in the determination of configuration averages [77], However, these correlations can be broken if the update of B is not done each step, but with a lower updating frequency. This is analogous to other approximately Markovian procedures employed in MC simulations (e.g., update of the maximum displacements allowed for individual atoms [78]). 

For a system with many degrees of freedom the canonical distribution of energy closely approximates the microcanonical distribution about the most... 

For this value of the energy the exponential factor becomes a constant and the distribution a function of H only, like the microcanonical ensemble. As a matter of fact, as the number of systems in the ensemble approaches infinity, the canonical distribution becomes increasingly sharp, thus approaching a microcanonical surface ensemble. 

Another way of looking at the Boltzmann distribution assumes that the energy spectrum consists of closely spaced, but fixed energy levels, e . The probability that level u is populated is specified in terms of the canonical distribution... 

In view of the ergodic hypothesis the average value of an observable property may be regarded as the quantity measured under specified conditions. In this way the internal energy of a system corresponds to the average energy of the canonical distribution ... 

Usually when closed, isothermal systems (N,V,T) are studied, the canonical distribution function is chosen ... 

Expression (21) tells us that if II is the canonical distribution function (7), given that... 

This dynamical formulation of the equilibrium correlations in an interacting system will be the starting point of our analysis of equilibrium electrolytes. Of course, this method gives results analogous to the more usual methods based on the canonical distribution 40... 

Consider the example of condensed phase transitions between vibrational states, which have energies that are significantly drfferent compared with knT. The momentum on the initial surface before a hop and the final surface momentum after the hop are considerably drfferent for typical values of the initial momentum sampled Irom a canonical distribution. This causes the two branches of the combined trajectory to quickly diverge, and action for the combined trajectory to grow rapidly. The result is that the integrand converges very quickly as a function of x, particularly after the and Fj integrations have been performed. 

We obtain the unperturbed canonical distribution. This is a special case of... 

The corresponding formulation was made by von Neumann2 for quantum mechanics. This entropy-maximizing (or information-minimizing) principle is the most direct path to the canonical distribution and thus to the whole equilibrium theory. It is understood that the extremalizing is conditional, i.e., certain expected values, such as that of the Hamiltonian, are fixed. 

If these ideas are combined with the concept of a stable distribution, i.e., one which is but minimally altered by any slight changes at the boundary, one is led to seek the S for which these effects have operated to their fullest extent, i.e., to the one which minimizes /[J ] among all those that have the same total expected mass, momentum, and energy. This of course is the method of obtaining the canonical distribution noted in Section I. 

Actually, this I(S) can be expressed explicitly in terms of an informationminimizing 3F which, as will be shown in Section VII, can be found as a locally canonical distribution (cf. the zero th approximation in the Chapman-Enscog process). We will denote it by E it is a functional of p, m , E and a point function of (x) on... 

As the next stage beyond the zero fluxes and stability underlying the canonical distribution, we consider more general initial (t = 0) basic macroscopic data (p, uE) and their basic macroscopic fluxes (t 0) on the boundary of Y, but under restrictions which may be described intuitively as follows Y is of reasonable shape and proportions—not much more irregular than a potato. The initial data (p, ux, E) vary slowly (percentagewise) with (x). On the boundary of Y the flux of p is zero those of ua and E are piecewise slowly varying (percentagewise) with position and time. 

---