# NVT/NPT Monte Carlo

1. Jun 23, 2013

### pyscho

Hello,

I am working on a MC simulation of a solid using NVT ensemble with Metropolis algorithm. The botlzman factor is of the form exp(-βΔU(r1,r2...rN)), where U is just the potential energy.

From thermodyanmics, dU=dq+dw (dq is the amount of heat put in, and dw is the amount of work done on the system). Since there is no work done (constant volume). dU=dq. Therefore this factor makes sense to me.

Now what I would like to do is let the volume change under constant pressure and temperature. That requires NPT ensemble. Again from thermodynamics du=dq+dw, but since volume is changing, dU=dq-pdV=dq-d(pV), thus d(U+pV)=dq. That's the enthalpy. What I would expect the boltzman factor to be is exp(-β(ΔU+pΔV)). Am I thinking wrong? However, I found it to be in the form exp(-β(ΔU+pΔV-NkTln(V'/V)) where V'=V+ΔV. Where does this extra term come from!?

Thank you very much for your help.

2. Jun 25, 2013

### moj20062001

I think,in this case dU has been take into two parts:
1.ΔU;that is an internal interaction change of energy,and
2.PdV-TdS;that the exra term which u see is TdS=nkTln(V'/V)
maybe!!!!!?!