Equilibrium probability distribution

The question of whether there exists an equilibrium probability distribution poo is difficult to answer in general. However, there is one simple scenario for which the answer is easy. Namely, when the system satisfies a condition called detailed balance. 

These two additional properties of the H in (5.6), together with its mono-tonic decrease, has led to its identification with the entropy defined by the second law of thermodynamics. It must be realized, however, that H is a functional of a non-equilibrium probability distribution, whereas the thermodynamic entropy is a quantity defined for thermodynamic equilibrium states. The present entropy is therefore a generalization of the thermodynamic entropy the generalized entropy is... 

The equilibrium probability distributions for the relative speed and the relative kinetic energy are frequently used in various one- or two-dimensional models. 

One can assume that each subchain is also sufficiently long, so that it can be described in the same way as the entire macromolecule, in particular, one can introduce the end-to-end distance for a separate subchain b2. The equilibrium probability distribution for the positions of all the particles in the... 

Pjq(.4,b) is the equilibrium probability distribution. If we define the scalar product (5 C) between two general variables B and C as... 

In statistical mechanics we do not measure observables directly. Instead we observe an average over all possible values. The averaging is done by means of a probability distribution function, which in classical mechanics is averaged over all of phase space. Let us compare an equilibrium ensemble with grand potential Q, and an arbitrary nearby ensemble prepared by a small perturbation, AQ. Let the equilibrium probability distribution function be f and that for the nearby... 

If W cannot be decomposed into block form then the system has a uniquely defined equilibrium state, P , for which ( fP/ ft) p=peq = 0, that is, W has a single zero eigenvalue whose eigenvector is the equilibrium probability distribution, W is asymmetric, but can be symmetrized using the condition of detailed balance at equilibrium,... 

Atomistic simulations provide the positions (and in the case of MD, velocities) of aU the atoms in the system consistent with the equilibrium probability distribution... 

Since transition probabilities are normalized, the denominator on the right-hand side of this equation is just the equilibrium probability distribution for configuration r. The parameter pg(r, Q is the probability that a system with initial configuration r will reside in state B at time t. The analogous committor for state A, pjr, g is similarly defined. If the timescale of molecular fiuctuations, is well separated from the typical time between spontaneous transitions, then p and pg will be nearly independent of t, for t <,i < "Trxn- this time regime, a trajectory will... 

Fig. 6.1 Concept of a thermodynamic reservoir. The thermodynamic state of the large system (resolved system plus reservoir) is defined by various macroscopic parameters that are assumed to be constant. This results in an equilibrium probability distribution for the small system, i.e. a thermodynamic state...

Two conclusions can be drawn from this result. First, the entanglement field constructed in this way does not affect the equilibrium probability distribution of the polymer chain. Second, P(Ro...Rn. o. .Xz.i) does not depend on a, which means that Xj must obey ideal gas statistits. In particirlar, the monomer distances between two entanglements - A j must be exponentially distributed. 

A macrostate of a macromolecule can always be described with the help of the end-to-end distance R ). To give a more detailed description of the macromolecule, one should use a method introduced by the pioneering work reported by Kargin and Slonimskii [1] and by Rouse [2] whereby the macromolecule is divided into N subchains of length M/N. The points at which the subchains join to form the entire chain (the beads) will be labelled 0 to N respectively and their positions will be represented by r°,r, ..., r. If one assumes that each subchain is also sufficiently long, and can be described in the same way as the entire chain, then the equilibrium probability distribution for the positions of all the particles in the macromolecule is determined by the multiplication of N distribution functions of the type of (5)... 

This is exactly the same form as the equilibrium probability distribution for the fluctuations in temperature (2) at the equilibrium temperature Ts = Tequ-For an ideal gas we have E = CyT and the fluctuations are in the Gaussian form in energy... 

For magnetic resonance relaxation (L = 2), the reorientational correlation functions [Eq. (7.20)] may be evaluated for a spherical molecule (n = n = 0) with the equilibrium probability distribution /(fio) = 1/47t. Since the molecules are assumed to reorient isotropically, all correlation functions decay exponentially with a single correlation time... 

We calculated the free energies of all the minima in order to determine the equilibrium probability distribution (see Section IV.C.2). We found that the several hundred lowest free energy minima have about the same free energy, and that no single minimum has an equilibrium occupation probability which exceeds 0.004. This is in stark contrast with unsolvated tetra-alanine, where the ground state had an equilibrium occupation probability of 0.748, and the lowest three potential energy states accounted for 0.936 of the total equilibrium probability. 

Once an adequate sample of minima and transition states has been found, we begin the dynamical analysis. Connectivity between minima and transition states has already been determined by the triples calculation (i.e., downhill searches). The free energy of each stationary point is calculated (using the vibrational frequencies), and from that the transition rates may be calculated. Then we can construct a Cv vs. T plot, determine equilibrium probability distributions, solve the Master equation, constmct the rate disconnectivity graph, and perform a full pathway analysis. 

One of the eigenvalues is zero, which corresponds to the equilibrium probability distribution (x = oo). The remaining eigenvalues will be negative. 

With this, we can figure out what the equilibrium probability distribution is when the system temperature, volume, and number of particles are constrained. All we need to do is to find the probability distribution that minimizes Eq. (2.16), subject to the normalization condition ... 

In the primitive MC method, the probability of making a transition from one conformation, x, to another, x, consists of two parts. The first is the probability of making a trial move from conformation x to conformation xl which is T(x x ). The second is the probability of accepting the trial move A x-> . r ) The underlying demand that the system should sample the equilibrium probability distribution of being in a given conformation x with probability p(x) is satisfied by the... 

Andricioaei and Straub have recently employed a similar acceptance probability where the trial step is sampled from a distribution function of a form proposed by Tsallis. In Tsallis statistics , the standard Gibbs entropy S = —k / dx p(x) In p(x) is modified to the form Sq = k J dx(l — Pq xf)j q — 1) which is equal to the Gibbs entropy formula in the limit that q =. The equilibrium probability distribution functions take the form... 

In this section we summarize methods for solution of the master equation, which couples the collisional relaxation of the highly excited unimolecular species with the microcanonical dissociation rates to determine, for a given temperature and pressure, the non-equilibrium probability distribution for the molecular population over energies and angular momenta, and thence the thermal rate coefficient k(T, P). The separability of molecular interactions in the gas phase into unimolecular events and bimolecular events enables the overall thermal dissociation process to be modeled by the two-dimensional master equation, expressed in continuum notation as... 

In statistical mechanics the goal is to generate a chain of microscopic states that sample the limiting, equilibrium probability distribution in phase space. Of course, the transition matrix is immense. For example, for NVE ensembles the transition matrix is an matrix. This is incomprehensibly large. In addition, the matrix elements are unknowable, since it is not possible to determine the transition probability between any two arbitrary points in phase space. 

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