Thermodynamic integration is a conceptually simple, albeit expensive, way to calculate free energy differences from MC or MD simulations. In this example, we will consider the calculation (again) of chemical potential in a LennardJones fluid at a given temperature and density, a task performed very well already by the Widom method (so long as the densities are not too high.) More details of the method can be found in Reference [15].
We begin with the relation derived in the book for a free energy
difference,
, between two systems which are identical
(same number of particles, density, temperature, etc.) except
that they obey two different potentials. System I obeys
and System II
. To measure this
free energy difference, we must integrate along a reversible path from
I to II. So let us suppose that we can write a ``metapotential'' that
uses a switching parameter, , to measure distance along this
path. So, when , we are in System I, and when we
are in System II. One way we might encode this (though this is not
necessarily a general splitting, as we shall see below) is
(210) 
Let us consider the canonical partition function for a system obeying
a general potential
:
(211) 
(212)  
(213)  
(214) 
The free energy difference between I and II is given by:
(215) 
To compute , we imagine two systems: System I has
``real'' particles, and 1 ideal gas particle, and system II has
real particles. The two free energies can be written:
(216)  
(217) 
For large values of , we see that
. So, we have another route to compute .
First, we tag a particle , call it the
``particle'', and apply the following modified potential to
its pairwise interactions:
(219) 
Next, we conduct many independent MC simulations at various values of and a given value of and , generating for each a table of vs. which can be integrated to yield a single value for . This turns out to be an expensive way to compute the chemical potential for a LennardJones fluid, compared to the Widom method (Sec. 6.1), for at least low to moderate densities.
I have done a rough comparison of the thermodynamic integration method described above to the grand canonical MC simulation technique described in Sec. 5.1. Below is a plot of vs. for three densities . Each point is computed from a single MC simulation using the code mclj_ti.c. The temperature was = 2.0, and run for 10 cycles for = 216. We see that the data is not terribly smooth; it is not clear how many more cycles would result in smoother data.


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