Adiabatic Process
絕熱過程是用來理解卡諾機的步驟，絕熱方程為PV^gamma=常數，其斜率較等溫過程為陡，而gamma值為Cp/Cv，為等壓比熱與等積比熱的比值，其值大於1。
可參考Adiabatic Process，上頭有計算的範例。
Partition Function
The partition function Z (sometimes denoted ) of a set of particles with energies for i = 1, ..., r is given by
where is thermodynamic beta.
Given a set of partition functions , the total partition function is their product,
Therefore, for N identical particles each with partition function.
Ising Model
This entry contributed by S. T. Wierzchon
A simple model used in statistical mechanics. The Ising model tries to imitate behaviour in which individual elements (e.g., atoms, animals, protein folds, biological membrane, social behavior, etc.) modify their behavior so as to conform to the behavior of other individuals in their vicinity. The Ising model has more recently been used to model phase separation in binary alloys and spin glasses. In biology, it can model neural networks, flocking birds, or beating heart cells. It can also be applied in sociology. More than 12,000 papers have been published between 1969 and 1997 using the Ising model.
This Ising model was proposed in the 1924 doctoral thesis of Ernst Ising, a student of W. Lenz. Ising tried to explain certain empirically observed facts about ferromagnetic materials using a model of proposed by Lenz (1920). It was referred to in Heisenberg's (1928) paper which used the exchange mechanism to describe ferromagnetism. The name became well-established with the publication of a paper by Peierls (1936), which gave a non-rigorous proof that spontaneous magnetization must exist. A breakthrough occurred when it was shown that a matrix Eric Weisstein's World of Math formulation of the model allows the partition function to be related to the largest eigenvalue Eric Weisstein's World of Math of the matrix Eric Weisstein's World of Math (Kramers and Wannier 1941, Montroll 1941, 1942, Kubo 1943). Kramers and Wannier (1941) calculated the Curie temperature using a two-dimensional Ising model, and a complete analytic solution was subsequently given by Onsager (1944).
To be more concrete, consider a set of N individuals arranged in a lattice. Each individual can be in one of two different states, say and -1. Let S be the space of all sequences or configurations
where or -1. Further, we define a function
where represents the energy of interaction between two neighbors in the lattice, k is Boltzmann's constant, and T stands for the temperature of the system (in K). The probability of a configuration s is defined now as follows:
where Z is the partition function
Assuming that each individual can be in one of q (where q > 2) states gives the potts model.
The computation of the partition function is a very hard problem. However, V. F. R. Jones observed an amazing connection between this problem and knot theory. Eric Weisstein's World of Math He showed that for the Ising model, the partition function yields the Arf invariant.
