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Thermodynamic integration is a technique for computing a difference in values of the free energy in two states characterized by different values of a parameter. This free energy difference is computed by numerically integrating the derivative of the free energy $ F $ with respect to the relevant model parameter.

The free energy $ F $ of a field theoretic model is given by the usual relationship

\[ F = -\ln Z \]

in which $ Z $ is a field theoretic partition function. Here and hereafter, we use "thermal" units for energy, in which $ kT = 1 $. For a theory that uses a partial saddle-point approximation,

\[ Z = \frac{1}{N} \int \! {\cal D}W_{-} \; e^{-H[W_{-},W_{+}^{*}] } \quad, \]

where $ N $ is a constant that is independent of the field configuration., Here, we use $ W_{-} $ to denote a list of the $ M -1 $ fields $ W_{1}, \ldots W_{M-1} $ that couple to composition fluctuations, and $ W_{+}^{*} $ to denote the partial saddle-point configuration of the pressure-like field $ W_{+} $$. Correspondingly, we use $ \int {\cal D}W_{-} $ to denote a functional integral with respect to these $ M - 1 $ field components.

Normalization constant

PSCF uses a convention for the constant $ N $ which is given for AB systems ( $ M = 2 $) by

\[ N \equiv \int {\cal D} W_{-} \exp \left \{ - \int \! d{\bf r} \; \frac{ W_{-}^{2}({\bf r}) }{v \chi } \right \} \]

When approximated by a product of Gaussian integrals on a lattice containing $ G $ mesh points in a volume $ V $, this yields

\[ \ln N = \frac{G}{2} \ln \left ( \frac{\pi \chi v G }{V} \right ) \quad. \]

The generalization to the case in which $ M > 2 $, but in which all $ M - 1 $ nonzero eigenvalues of the projected $ \chi $ matrix are negative yields a constant

\[ \ln N = \frac{G}{2} \sum_{\alpha = 1}^{M-1} \ln \left ( \frac{2\pi |\lambda_{\alpha}| v G }{ MV } \right ) \quad. \]

Here, $ \lambda_{1}, \dots, \lambda_{M-1} $ denote the first $ M - 1 $ eigenvalues of the projected $ \chi $ matrix, excluding the eigenvalue $ \lambda_{M} $ that is zero by construction.

Note that this above convention for $ N $ yields a value that depends on the Flory-Huggins interaction parameters, the monomer volume $ v $, and system volume $ V $. These dependencies must be taken into account when computing derivatives of $ F $ with respect to any of these variables.

Free energy derivatives

Suppose $ x $ is a real-valued parameter that appears in the field theoretic Hamiltonian $ H $ or the above exprssion for $ N $, or both. The thermodynamic integration method is based on a straightforward theorem that states that the derivative of free energy $ F $ with respect to $ x $ is given by a sum

\[ \frac{\partial F}{\partial x} = \left \langle \frac{\partial H}{\partial x} \right \rangle + \frac{\partial \ln N}{\partial x} \quad. \]

Here, $ \partial H/\partial x $ is a derivative evaluated in a fixed field configuration, and $ \langle \cdots \rangle $ denotes an equilibrium ensemble average evaluated using a Hamiltonian with the specified value of parameter $ x $. To simplify notation, it is sometimes useful to express this an average

\[ \frac{\partial F}{\partial x} = \langle \Psi \rangle \]

in which we define a functional of the fields

\[ \Psi = \frac{\partial H}{\partial x} + \frac{\partial \ln N}{\partial x} \]

that contains a field-indepdent term arising form the derivative of $ \ln N $.

Numerical integration

The change in $ F $ associated with a change in $ x $ from an initial value $ x_{0} $ to a final value $ x_{1} $ is obtained by integrating the derivative $ \partial F/ \partial x $ over this range. This sort of integral can be computed numerically by either of two methods:

Separate simulations : Equilibrium simulations can be performed at a set of values of $ x $ within the range of interest, each of which yields a value of $ \partial F/\partial x $ at a single value as an ensemble average. The required integral is then approximated by numerical integration of the resulting sequence of values.
Continuous integration : The integral may be computed by using a ramp to continuously vary the parameter of interest over the course of a simulation.

Continuous integration may be used either to compute the change in free energy associated with a change in a parameter of the standard field theoretic Hamiltonian (such as an interaction parameter or the monomer concentration) or as a change in free energy associated with changes in the strength of a perturbation that is added to the standard Hamiltonian to create an unphysical perturbed Hamiltonian (as done in the Einstein crystal integration method).

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