Probability density in phase space

In this chapter, we present an illustrative calculation of the probability density in phase space and the partition function of the canonical ensemble. We then again derive ide gas thermodynamics. 

In order for the temperature of a system S to remain constant, the system must be in thermal contact with a heat bath B (Fig. 5.1). Assume that the system has energy Es and the heat bath has energy Eb. The composite system is isolated with a constant total energy E = Eb + Es. The bath can be arbitrarily large. Consequently, we may assume Eb Es. 

Since the system and the heat bath are in thermal contact, their temperatures will be the same at equilibrium, 

Energy can flow freely between the system and the heat bath. Consequently is no longer fixed, but can fluctuate. 

Assume that system S can attain any one of a large number of different energy levels Ei,i = 1,2. with probability ps(Ei). For each energy level Ei, the system can be in any of Q(Ei) microscopic states. We will use a counter j = 1,2. S2(Ei) for each of the microstates at each energy level Ej. 

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For particles interacting through the potential l/(r ), the canonical-ensemble probability density in phase space is... 

The equilibrium probability density in phase space for a system with total energy Eq is therefore given by ... 

Instead of describing the stochastic dynamics via a Langevin equation, we use the Fokker-Planck equation, which is the evolution equation for the probability density in phase space. For an Al-particle system. 

Distribution function and probability density in phase space... 

This Wigner representation of the density pw q, p) proves particularly useful since it, satisfies a number of properties that are similar to the classical phase-space distribu tion pd(q, p). For example, if p = pure state, then fdppw = probability density in coor- dinate space. Similarly, integrating pw over q gives the probability density in ( momentum space. These features are shared by the classical density p p, q) in phase space. Note, however, that pw is not a probability density, as evidenced by if the fact that it can be negative, a reflection of quantum features of the dynamics, ) [165], 3... 

An example where the smooth probability density in Euclidean space is not quite the right one is in the setting of conservative (Hamiltonian) systems such as our N-body molecular system, since the evolution is restricted by invariants. The most obvious of these is the energy which we know to be a constant of motion. Therefore we need to work not on open subsets of the phase space R" of our differential equations, but on lower dimensional submanifolds embedded within the phase space, e.g. the energy surface. It will be necessary to assume a density that is defined over the submanifold of constant energy. If other invariants are present, such as fixed total momentum, the discussion would need to be modified to reflect this fact. 

In a canonical ensemble the probability f canon(T E) of visiting a point in phase space wi an energy E is proportional to the Boltzmarm factor, = exp(—E/lcgT), multiplied by tl density of states, (E), where the number of states between E and E + dE is given 1 n E)6E. Thus ... 

The probability density p(p, q, t) in phase space satisfies the continuity equation... 

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

Consider a gas whose phase density in T space is represented by a microcanonical ensemble. Let it consist of molecules with //-spaces pi with probability distributions gt. Denote the element of extension in pi by fa. Since energy exchanges may occur between the molecules, pi cannot be represented by a microcanonical distribution. There must be a finite density corresponding to points of the ensemble that do not satisfy the requirement of constant energy. Nevertheless, the simultaneous probability that molecule 1 be within element di of its p-space, molecule 2 within dfa of its //-space, etc., equals the probability that the whole gas be in the element... 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

The interpretation of any distribution function as a probability density function in phase space leads to the requirement... 

Lutz also compared his results with those predicted by the fractional Klein-Kramers equation for the probability density function/(x, v, f) in phase space for the inertia-corrected one-dimensional translational Brownian motion in a potential Eof Barkai and Silbey [30], which in the present context is... 

Considering an ensemble of initial conditions each representing a possible state of the system, we express the probability of a given ensemble or density distribution of system points in phase space T by a continuous function... 

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