Master Equation

This is the first post on a series on Monte-Carlo Methods. We’ll explore the traditional way in which they were used and also more recent variations and adaptations.

Named after a casino in Monaco, Monte-Carlo methods work in a way that closely mimics the gambling that occurs there. Dice are rolled, and fortune decides who is the winner for the day. Players hope they can beat the odds and make their dreams come true., while casino owners rest assured in the knowledge that the large fluctuations that attract gamblers will average out and leave them with a healthy profit.

Monte-Carlo methods are, in a way, a formalization of this process in a way that allows them to be used to model stochastic problems. Despite it’s variety, they always involve generating random configurations of the system (rolling the dice), measuring some property of this configuration (who is the winner), and, finally, averaging over many such trials (the overall profit).

Physical processes are often stochastic and just as often we don’t posses detailed information about the system. What will be the result of a coin toss? Where exactly are the impurities in a semi conductor? However, we must somehow account for this while studying these problems, and this is where Monte-Carlo methods have proven most useful.

Stochastic models where the transitions from a given state s to another state s’ occur spontaneously at a rate [tex]\omega_{s\to s’}\ge0[/tex], have a probability distribution [tex]P\left(s,t\right)[/tex], of being in state s at time t that evolves deterministically in time according to a master equation of the form:

[tex]\displaystyle\frac{\partial}{\partial t}P\left(s,t\right)=\sum_{s’}\omega_{s’\to s}P\left(s’,t\right)-\sum_{s}\omega_{s\to s’}P\left(s,t\right)[/tex]

where the first summation accounts for transitions from all states s’ in to state s and the second term accounts from transitions away from state s on to some other state s’. This equation describes the probability flow that is responsible from creating or destroying any given configuration. Both terms are correlated in such a way that the usual normalization condition [tex]\sum_{s}P\left(s’,t\right)\equiv1[/tex] is valid at all possible values of t. If we know the complete [tex]P\left(s,t\right)[/tex] at any point in time, we can use this equation to determine the form of [tex]P\left(s,t\right)[/tex] at any other instant. One important consequence of this fact is that processes of this type don’t posses any type of memory about how they reached state s, thus clearly marking them as Markov Chain processes. It is also important to note that the [tex]\omega_{s\to s’}[/tex] are rates and not probabilities, being able to take on values larger than unity and defining an intrinsic time scale which can be modified by rescaling all the [tex]\omega[/tex]s by a constant.

Given that the equation above describes a time dependent process, we can follow the usual physical definition of equilibrium, namely, that an equilibrium process is such that any explicit time dependence is eliminated, or, mathematically, the point at which:

[tex]\displaystyle\frac{\partial}{\partial t}P\left(s,t\right)\equiv0[/tex]

from where we can easily see that at equilibrium, the system must obey:

[tex]\displaystyle\sum_{s’}\omega_{s’\to s}P\left(s’\right)=\sum_{s}\omega_{s\to s’}P\left(s\right)[/tex]

As a particular case, we have:

[tex]\displaystyle\omega_{s’\to s}P\left(s’\right)=\omega_{s\to s’}P\left(s\right)[/tex]

This condition is usually referred to as the ”detailed balance condition”. One should note, that detailed balance is a ‘’sufficient” condition for equilibrium, but not a required condition. This means that some system can disobey this principle and still be at equilibrium, but it also means that any process that obeys this prescription is necessarily in equilibrium.

In the next post in this series, we’ll start the discussion of the different ways in which we can sample configuration space. In the mean time, I would love to hear any comments or suggestions you might have.

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