phys212ln14 - Physics 212: Statistical mechanics II, Fall...

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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2005 Lecture XIV An important difference between classical and quantum mechanics is that, in quantum me- chanics, the Hamiltonian determines both the dynamic and the static properties. The same Hamiltonian H appears in both the density matrix e- H and the Schrodinger equation of mo- tion. In classical mechanics the situation is quite different. For instance, given the Ising model Hamiltonian, H =- J X h ij i s i s j , (1) it is not clear how the system will evolve under time. An even harder question is to understand how such a system will approach thermal equilibrium. (For classical Hamiltonian systems the time evolution is specified, but often we have dealt with effective models like the Ising model for which the underlying Hamiltonian system is not clear.) We would like to know, given a spin configuration (not necessarily the ground state) for the Ising model, how the configuration evolves in time. It turns out that there are many different answers to this question, and that the correct answer depends on the system to be modeled. One property that should be satisfied is that the thermal equilibrium configuration be preserved by the dynamics. This reflects a fundamental thermodynamic assumption: there is a static probability distribution, given by the Boltzmann factor, over the states of the system. Note that in spirit this is much like our work on understanding the increase of entropy in a dilute gas: we want to understand how microscopic dynamics are related to macroscopic thermodynamic principles. The idea of a master equation is that the evolution of the probability distribution at time t + dt should be memoryless (not depend on times prior to t ) and determined by the probability distribution at time t . Let a different configurations of the spin system be denoted { s } , which stands for specific values for all the different spins. Starting at time t from a configuration { s } , we will assume for the moment that over a short time interval dt at most one spin of the system is flipped (that is, if spin flips occur independently, then in a very short time interval the probability of flipping flipped two spins will be much smaller than the probability of flipping one spin.). Let w j ( { s } ) dt be the probability that spin j is flipped in a small increment of time. Then we can write, for the evolution of the probability distribution P ( { s } ; t ), dP...
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This note was uploaded on 08/01/2008 for the course PHYSICS 212 taught by Professor Moore during the Fall '06 term at University of California, Berkeley.

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phys212ln14 - Physics 212: Statistical mechanics II, Fall...

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