Probability density function , equilibrium

Janicka, J., W. Kolbe, and W. Kollmann (1979). Closure of the transport equation for the probability density function of turbulent scalar fields. Journal of Non-Equilibrium Thermodynamics 4, 47-66. 

A quantity relevant to the discussion of the relaxation in the presence of a sink is the broadness of the probability density function of the initial condition. The following case where the system is made reactive (knr = 0) at times t 0, but is unreactive (/cnr = 0) for t < 0—corresponding to physical situations—can be solved easily. Indeed, the initial distribution p0(x0) equals the Boltzmann equilibrium distribution [see also Eq. (4.157)], that is,... 

The statistical behavior of interest is encapsulated in the equilibrium probability density function P )( q c). This PDF is determined by an appropriate ensemble-dependent, dimensionless [6] configurational energy 6( q, c). The relationship takes the generic form... 

Equation (10.40) is sometimes found in a simpler form at the cost of hiding the complexity of the terms involved. This form is based on the introduction of a probability density function for the reaction coordinate and the associated potential of mean force, in contrast to previously, where we considered the probability density of a particular arrangement of n atoms. Let I l(Q )d,Q be the probability of finding the reaction coordinate in the range Q, Q + dQ. Then, from equilibrium statistical mechanics (see Appendix A.2), the probability density function I l(Q ) is given by... 

Equation (42) cannot be used if NO concentrations approach their equilibrium values, since the net production rate then depends on the concentration of NO, thereby bringing bivariate probability-density functions into equation (40). Also, if reactions involving nitrogen in fuel molecules are important, then much more involved considerations of chemical kinetics are needed. Processes of soot production similarly introduce complicated chemical kinetics. However, it may be possible to characterize these complex processes in terms of a small number of rate processes, with rates dependent on concentrations of major species and temperature, in such a way that a function w (Z) can be identified for soot production. Rates of soot-particle production in turbulent diffusion flames would then readily be calculable, but in regions where soot-particle growth or burnup is important as well, it would appear that at least a bivariate probability-density function should be considered in attempting to calculate the net rate of change of soot concentration. 

These formulas show that knowledge of the average production rates of near-equilibrium species and of heat relies on knowledge of the joint probability-density function for Z and x Alternatively, since P is a function... 

An alternative way of relating concentrations (mass fractions) of individual species to/ is the assumption of chemical equilibrium. An algorithm based on minimization of Gibbs free energy to compute mole fractions of individual species from / has been discussed by Kuo (1986). The equilibrium model is useful for predicting the formation of intermediate species. If such knowledge of intermediate species is not needed, the much simpler approximation of mixed-is-burnt can be used to relate individual species concentrations with/. In order to calculate the time-averaged values of species concentrations the probability density function (PDF) approach is used. 

In modem physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity... 

We also need some background material about (19). If m(x) denotes the equilibrium probability density function of x(t), i.e. the probability density to find a trajectory (reactive or not) at position x at time t, m(x) satisfies the (steady) forward Kolmogorov equation (also known as Fokker-Planck equation)... 

We suppose that a small probing held Fj, having been applied to the assembly of dipoles in the distant past (f = —oo) so that equilibrium conditions have been attained at time t = 0, is switched off at t = 0. Our starting point is the fractional Smoluchowski equation (172) for the evolution of the probability density function W(i), cp, t) for normal diffusion of dipole moment orientations on the unit sphere in configuration space (d and (p are the polar and azimuthal angles of the dipole, respectively), where the Fokker-Planck operator LFP for normal rotational diffusion in Eq. (8) is given by l j p — l j /> T L where... 

A-dimensional phase space). A probability density function / /-, / ) characterizes the equilibrium state of the system, so that is the probability... 

Here pj = pj +pjy + pj7 and U is the potential associated with the inter-particle interaction. The function/(/ , p ) is an example of a joint probability density function (see below). The staicture of the Hamiltonian (1.184) implies that f can be factorized into a term that depends only on the particles positions and terms that depend only on their momenta. This implies, as explained below, that at equilibrium positions and momenta are statistically independent. In fact, Eqs (1.183) and (1.184) imply that individual particle momenta are also statistically independent and so are the different cartesian components of the momenta of each particle. 

The equilibrium statistical mechanics of the fluids can be discussed in terms of the n-particle probability density function p(l 2 n) associated with finding a particle centered at ri with orientation o>i, another particle at t2 with orientation <02, etc. we shall in addition often work with the nondimensional functions... 

Thermodynamic fluctuation theory characterises equilibrium fluctuation by the so-called Einstein relation connecting the probability density function g with the (appropriately defined) entropy function S ... 

In the next chapter, we will consider the nonequilibrium behavior of matter in the most general way by deriving the spatial and temporal variations in density, average velocity, internal and kinetic energy, and entropy. We will use the formal definitions of these quantities introduced in this chapter, including the possibility of their spatial and temporal variations via the probability density function described by the full Liouville equation. In the next chapter, we will also formally define local equilibrium behavior and look at some specific, well-known examples of such behavior in science and engineering. 

Both authors argued that for dynamically similar breakage mechanisms, the equilibrium DSD should only depend on the ratio of disruptive (Xc) to cohesive (Xs and/or xj) forces acting on the drops. Thus, the individual DSDs could be collapsed to a single correlation by normalization with d 2- Defining X = d/d32, the volume probability density function becomes... 

Consider two different instantaneous configurations of a system, namely and r. The probability of finding the system in any of these two configurations is dictated by equation (1). Consider also proposing some arbitrary transition scheme to go from configuration r to configuration r. The probability density function that the evolution of a system known to be at will bring it near is denoted by K r r ). Note that K could be an actual model for the kinetics of a process or a mathematical abstraction. At equilibrium, the system should be as likely to move from r to as in the reverse direction. This is stated by the condition of detailed balance, which can be written as... 

A-dimensional phase space). A probability density function/(r, /A) characterizes the equilibrium state of the system, so that/,p )dr dp is the probability to find the system in the neighborhood dr dp = dr i,.dpN of the corresponding phase point. In a canonical ensemble of equilibrium systems characterized by a temperature T the function/ r, p ) is given by... 

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