# Are there some necessary and sufficient conditions that a system can be modeled using Monte Carlo Methods?

Is there a certain set of conditions a system need to follow such that it can be effectively modeled using any "general" Monte Carlo method, maybe to find some average value of a thermodynamic quantity.

• Monte-Carlo is one of those term to which different people give a different meaning. I reckon you need to be more specific here.
– user154997
Commented Oct 19, 2017 at 17:28
• Especially considering the two rather specific examples of hypothesis you gave!
– user154997
Commented Oct 19, 2017 at 17:29
• The answer to this question is not just a function of the system to be modeled, but of what information you want to learn from the system through the model.
– Paul
Commented Oct 19, 2017 at 17:34
• In Metropolis & Ulam's early paper (J. American Statistical Association, Volume 44 Number 247 p335-341 (1949)), the abstract includes "The method is, essentially, a statistical approach to the study of differential equations, or more generally, of integro differential equations that occur in various branches of the natural sciences." One does not find mention of either of you putative conditions as a limit in their paper. Commented Oct 19, 2017 at 18:39
• @Paul Yes Monte Carlo can be used for various applications. If, I do not want to be very general, I would be more interested in average of some thermodynamic quantity. Commented Oct 19, 2017 at 18:45

The short answer is: No. There are no requirements on the system you want to model.

There are plenty of different flavors of Monte Carlo (MC) and the method you choose has to meet various conditions. The most common version of MC are Markov-chain Monte-Carlo methods such as Metropolis-Hastings. The two basic conditions are

• Ergodicity: You have to be able to reach all relevant parts of the phase space by a) sampling long enough and b) choosing your steps such that there are no "forbidden" areas in the phase space.

• The sampling weight for each state point has to converge to the Boltzmann distribution.

When detailed balance is fulfilled, you can prove that the Boltzmann distribution is a stationary (equilibrium) distribution. Technically, however, detailed balance is not a necessary condition.

To stress the first point: These are requirements on the Monte-Carlo method, not on the system. The most "general" Monte-Carlo method would be to sample random points in phase space and weigh them according to the Boltzmann distribution. This method tends to converge extremely slowly, but is not dependent on the conditions stated above.

• So Ergodicity is only needed for MCMC and Metropolis-Hastings, however it can be violated for some other Monte-Carlo Method? Commented Oct 21, 2017 at 5:30
• The term ergodicity is typically used for processes that evolve with time, such as a MCMC method exploring phase space. If you randomly sample points in phase space this is not the case. So, depending on your view, either the term ergodicity does not apply or ergodicity is trivially fulfilled. Commented Oct 25, 2017 at 5:26