By William Geraint V. Rosser
Advent to Statistical Physics (Mathematics and Its purposes) [Paperback
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Extra info for An introduction to statistical physics (1982)
Section 9 discusses a class of exactly solvable models comprising superficially very different systems, ranging from Glauber dynamics for the spin relaxation of the one-dimensional Ising model (Glauber, 1963) to branching and coalescing random walks which describe diffusion-limited pair reactions. Various relations between these and other models have been known for some time and can be found scattered in the literature cited below. 3 are devoted to bringing some order into this web of relations.
3) This equation can be obtained from the discrete-time description by taking the limit of continuous time, but is intuitively easy to understand directly from the definition of the process: the probability of finding the particle at site x increases through right (left) jumps from site x - 1 (x + 1) with rate Dn (DL). This assumes that the particle has not been at site x before the jump. On the other hand, if it was on site x, then the probability of finding it there decreases with rate DR + DL because it can hop in either direction away from site x.
However, the consequences of the quantum Hamiltonian formulation for these processes are largely unexplored and thus represent an area of future research. 1 17 Quantum Hamiltonian formalism for the master equation The master equation Our real concern is not the modelling of a physical system by some stochastic equation (the first approximation discussed in the introduction), but the analytical treatment of such an equation once it has been obtained through experimental observation and physical intuition.