Canonical equilibrium distribution

The Liouvillian iLo- = Ho, , where , is the Poisson bracket, describes the evolution governed by the bath Hamiltonian Hq in the held of the fixed Brownian particles. The angular brackets signify an average over a canonical equilibrium distribution of the bath particles with the two Brownian particles fixed at positions Ri and R2, ( -)0 = Z f drNdpNe liW J , where Zo is the partition function. 

The time derivative of the solvent polarization can be written as AE R) = VrAE R). The canonical equilibrium distribution is given by = Zq c Hw with Zo = f dRdP e Hw Equation (102) provides a well-... 

Equation [7] expresses the balance between the flux of all other states X toward X (the second term on the right-hand side of eqn [7]), leading to an inaease of P(X, t), and the flux out of the StateX (the first term on the right-hand side of eqn [7]), leading to a decrease of P(X, t). Now, for the application of the importance sampling MC method in statistical physics, one requires that the transition probability W(X X ) satisfy the detailed balance principle with the (canonic) equilibrium distribution Peq(X) = Z" exp(-V(X)/feB r), Z being the partition ftinaion... 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

The first factor on the right-hand side of the above equation, p(z0), is the distribution of initial conditions zo, which, in many cases, will just be the equilibrium distribution of the system. For a system at constant volume in contact with a heat bath at temperature T, for instance, the equilibrium distribution is the canonical one... 

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. 

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. 

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). 

One can do dynamics under this Hamiltonian by making the trajectory undergo an elastic reflection whenever it strikes one of the infinite barriers (14). Under H, the different parts of S would be visited with the same relative frequency as Tn an unconstrained equilibrium machine experiment, but with a much greater absolute frequency thereby allowing a representative sample of, say, 100 representative points on S to be assembled in a reasonable amount of computer time. If the equilibrium distribution is canonical the momentum distribution will be Maxwel1ian and independent of coordinates hence, representative points (p,q) can be generated by taking c[ from an equilibrium Monte... 

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 ... 

This is the most general form of the Liouvillian when one aims to describe the process by only two variables and their canonical distribution has to be recovered. The first term of is the Liouvillian given by Eq. (4.6) after performing the average, and the second term is added to satisfy the equilibrium distributions of /Iq and v. The rq>resentation of L g in terms of the creation and annihilation operators will turn out to be useful in Section V, where some properties of the Liouvillian (4.11) will be discussed. Note that... 

Kusaka I, Wang ZG, Seirrfeld JH (1998a) Direct evalrration of the equilibrium distribution of physical clusters by a grand canonical Monte Carlo simrrlation. J Chem Phys 108 3416-3423 Kusaka I, Wang ZG, Seinfeld JH (1998b) Binary nucleation of sulfuric acid-water Monte Carlo simulation. J Chem Phys 108 6829-6848... 

The continuous metadynamics algorithm can be applied to any system evolving under the action of a dynamics whose equilibrium distribution is canonical at an inverse temperature 1// . In a molecular dynamics scheme this requires that the evolution is carried out at constant temperature, by using a suitable thermostat [51]. In the continuous version of metad3mamics, Gaussians are added at every MD step and act directly on the microscopic variables. This generates at time t extra forces on x that can be written as... 

The explanation for this paradox is that a canonical or thermal system maintains the equilibrium distribution of internal energy states through collisions. Thus, Eq. (1.8) does not describe the time behavior of a canonical ensemble. What it docs describe is a... 

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).

In the limit where the number of configurations M generated tends to infinity, the distribution of states X obtained by this procedure is proportional to the equilibrium distribution Pe (X), provided there is no problem with the ergodicity of the algorithm (this point will be discussed later). Then, the canonical average of any observable Z(X) is approximated by a simple arithmetic average. 

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,... 

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)... 

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