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Why do we perform different trial moves in Monte Carlo simulations in Statistical Mechanics? For example, in NVT ensemble simulations, Why only atom displacement moves? In NPT ensemble, why do we need to perform both volume adjustment moves and atom displacement moves? What are the variables that we are minimizing ? What are our main objectives behind performing these trial moves?

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The question is basically about the Monte Carlo move set, and their choice.

The Monte Carlo method basically requires that the system diffuse in the "phase space". This is done by first suggesting a move and then calculating the probability of acceptance of such move. This is to say that the move set will have to "satisfy" the conditions of the simulation. If you are doing constant-N,V,T simulation, the condition of constant volume will of course preclude the changes in the volume. If you are doing the constant- N,p,T simulation, the volume has to be changing so that each sampled phase point have "same" pressure. In both NpT and NVT , the particles are neither destroyed or created.

The main objective of the any move in the move set is to allow for various "ways" for the system to move about and to (and into) new regions in phase space, also called sampling. For mono-atomic simulations (i.e. noble gases), it is natural to think of the simplest move: displacing each particle independently. This does not mean that other MC moves are not to be considered; of special interest are what are called "collective moves" where more than one atom is displaced together/nearly together. For example: if you are simulating water, then you will move all the three atoms (the two hydrogens and the oxygen) belonging to the same water molecule together. If not, most of the moves that you make will take the system to configurations having very high energy, and will be rejected; and the system will only diffuse very very slowly in phase space. For water system, it might be prudent to have a rotation move, where the molecule as a whole tumbles randomly.

What is the variable that is "minimised" ? This is tricky. The common lore is that the variable (in the move) be tuned so that 30-50% of the moves are accepted. For example, in the dispacement move, the variable is the maximum-value of such random displacement.

I would suggest the book by Allen and Tildesley "Computer Simulation of Liquids", Chapter 4 (Monte Carlo Methods)

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