stylistic:style_transfer, information:classification, linguistic:voice_change
Prior probability in statistical mechanics

While in Bayesian statistics the prior probability is used to represent initial beliefs about an uncertain parameter, in statistical mechanics the a priori probability is used to describe the initial state of a system. The classical version is defined as the ratio of the number of elementary events (e.g. the number of times a die is thrown) to the total number of events—and these considered purely deductively, i.e. without any experimenting. In the case of the die if we look at it on the table without throwing it, each elementary event is reasoned deductively to have the same probability—thus the probability of each outcome of an imaginary throwing of the (perfect) die or simply by counting the number of faces is 1/6. Each face of the die appears with equal probability—probability being a measure defined for each elementary event. The result is different if we throw the die twenty times and ask how many times (out of 20) the number 6 appears on the upper face. In this case time comes into play and we have a different type of probability depending on time or the number of times the die is thrown. On the other hand, the a priori probability is independent of time—you can look at the die on the table as long as you like without touching it and you deduce the probability for the number 6 to appear on the upper face is 1/6.

In statistical mechanics, e.g. that of a gas contained in a finite volume V {\displaystyle V}, both the spatial coordinates q i {\displaystyle q_{i}} and the momentum coordinates p i {\displaystyle p_{i}} of the individual gas elements (atoms or molecules) are finite in the phase space spanned by these coordinates. In analogy to the case of the die, the a priori probability is here (in the case of a continuum) proportional to the phase space volume element Δ q Δ p {\displaystyle \Delta q\Delta p} divided by h {\displaystyle h}, and is the number of standing waves (i.e. states) therein, where Δ q {\displaystyle \Delta q} is the range of the variable q {\displaystyle q} and Δ p {\displaystyle \Delta p} is the range of the variable p {\displaystyle p} (here for simplicity considered in one dimension). In 1 dimension (length L {\displaystyle L}) this number or statistical weight or a priori weighting is L Δ p / h {\displaystyle L\Delta p/h}. In customary 3 dimensions (volume V {\displaystyle V}) the corresponding number can be calculated to be V 4 π p 2 Δ p / h 3 {\displaystyle V4\pi p^{2}\Delta p/h^{3}}. In order to understand this quantity as giving a number of states in quantum (i.e. wave) mechanics, recall that in quantum mechanics every particle is associated with a matter wave which is the solution of a Schrödinger equation. In the case of free particles (of energy ϵ = p 2 / 2 m {\displaystyle \epsilon ={\bf {p}}^{2}/2m}) like those of a gas in a box of volume V = L 3 {\displaystyle V=L^{3}} such a matter wave is explicitly ψ ∝ sin (l π x / L) sin (m π y / L) sin (n π z / L), {\displaystyle \psi \propto \sin(l\pi x/L)\sin(m\pi y/L)\sin(n\pi z/L),} where l, m, n {\displaystyle l,m,n} are integers. The number of different (l, m, n) {\displaystyle (l,m,n)} values and hence states in the region between p, p + d p, p 2 = p 2, {\displaystyle p,p+dp,p^{2}={\bf {p}}^{2},} is then found to be the above expression V 4 π p 2 d p / h 3 {\displaystyle V4\pi p^{2}dp/h^{3}} by considering the area covered by these points. Moreover, in view of the uncertainty relation, which in 1 spatial dimension is Δ q Δ p ≥ h, {\displaystyle \Delta q\Delta p\geq h,} these states are indistinguishable (i.e. these states do not carry labels). An important consequence is a result known as Liouville's theorem, i.e. the time independence of this phase space volume element and thus of the a priori probability. A time dependence of this quantity would imply known information about the dynamics of the system, and hence would not be an a priori probability. Thus the region Ω:= Δ q Δ p ∫ Δ q Δ p, ∫ Δ q Δ p = c o n s t., {\displaystyle \Omega :={\frac {\Delta q\Delta p}{\int \Delta q\Delta p}},\;\;\;\int \Delta q\Delta p=\mathrm {const.},} when differentiated with respect to time t {\displaystyle t} yields zero (with the help of Hamilton's equations): The volume at time t {\displaystyle t} is the same as at time zero. One describes this also as conservation of information.

In the full quantum theory one has an analogous conservation law. In this case, the phase space region is replaced by a subspace of the space of states expressed in terms of a projection operator P {\displaystyle P}, and instead of the probability in phase space, one has the probability density Σ:= P Tr (P), N = Tr (P) = c o n s t., {\displaystyle \Sigma :={\frac {P}{{\text{Tr}}(P)}},\;\;\;N={\text{Tr}}(P)=\mathrm {const.},} where N {\displaystyle N} is the dimensionality of the subspace. The conservation law in this case is expressed by the unitarity of the S-matrix. In either case, the considerations assume a closed isolated system. This closed isolated system is a system with (1) a fixed energy E {\displaystyle E} and (2) a fixed number of particles N {\displaystyle N} in (c) a state of equilibrium. If one considers a huge number of replicas of this system, one obtains what is called a microcanonical ensemble. It is for this system that one postulates in quantum statistics the "fundamental postulate of equal a priori probabilities of an isolated system." This says that the isolated system in equilibrium occupies each of its accessible states with the same probability. This fundamental postulate therefore allows us to equate the a priori probability to the degeneracy of a system, i.e. to the number of different states with the same energy.