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In a field theoretic treatment of an incompressible system, the partial-saddle approximation (PSA) is an approximation for the treatment of the pressure like field that acts to constrain the total density. When it is applicable, the approximation yields a theory with real statistical weights for all allowed field configurations.

Standard AB systems ( \( M = 2 \))

We first consider the PSA for an incompressible standard AB system with \( \chi > 0 \). In the fully fluctuating theory for such a system, the partition function can be expressed as an functional integral over a real exchange field \( W_{-} \) and an imaginary pressure-like field \( W_{-}({\bf r}) = i\omega_{+} \).

In the PSA for such a system, \( Z \) is approximated by functional integral with respect to \( W_{-} \) in which the Hamiltonian \( H \) is evaluated in a configuration in which \( W_{+} \) is chosen so as to give a vanishing partial functional derivative of \( H \) with respect to \( W_{+}({\bf r}) \). This yields an approximate for the partition function as an integal

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

in which \( \int D W_{-} \) denotes a functional integral over the exchange field \( W_{-} \), while \( W_{+}^{*} \) denotes a configuration of \( W_{+}({\bf r}) \) that is chosen to satisfy the so-called partial-saddle point condition

\[ 0 = \left . \frac{ \delta H } { \delta W_{+}({\bf r}) } \right |_{W_{-}} = \frac{1}{v} \left [ \phi_{1}{\bf r}) + \phi_{2}({\bf r}) - 1\right ] \]

for all \( {\bf r} \). Note that this condition is satisfied if and only if \( \phi_{1}({\bf r}) + \phi_{2}({\bf r}) = 1 \). Partial saddle-point field configurations thus satisfy a mean-field version of the incompressibility constraint that is identical to the form of this constraint used in SCFT. SCFT solutions also satisfy the requirement that \( \delta H/\delta W_{-}({\bf r}) = 0 \), however, while PS-FTS methods allow \( W_{-}({\bf r}) \) to fluctuate.

Partial saddle-point field configuration for states characterized by a real-valued exchange field \( W_{-}({\bf r}) \) always have a real-valued field \( W_{+}({\bf r}) \), corresponding to a pure imaginary value of \( \omega_{+}({\bf r}) \). Such a field configurations always yield real monomer w-fields fields, given by

\[ w_{1}({\bf r}) = W^{*}_{+}({\bf r}) + W_{-}({\bf r}) \quad \]

\[ w_{2}({\bf r}) = W^{*}_{+}({\bf r}) - W_{-}({\bf r}) \quad. \]

Partial saddle-point field configurations are also characterized by real values for \( \ln Z_{\rm id}[w] \) and the overall Hamiltonian \( H \).

The partial saddle-point Hamiltonian for such a standard AB system can be expressed as a sum

\[ H = H_{\rm f} + H_{\rm id} \quad, \]

in which

\[ H_{\rm f} = \frac{1}{v} \int \! d{\bf r} \; \left \{ \frac{ W_{-}^{2} }{ \chi } - W_{+}^{*} + \frac{\chi}{4} \right \} \quad, \]

and

\[ \begin{align} H_{\rm id}[w] & = -\ln Z_{\rm id}[w] \\ & = \frac{V}{v}\sum_{a} \frac{\overline{\phi}_{a}}{N_{a}} \left [ \ln \left ( \frac{\overline{\phi}_{a}}{Q_{a}} \right ) - 1 \right ] \quad. \end{align} \]

Here, \( a \) is an index for a molecular species, while \( \overline{\phi}_{a} \), \( N_{\alpha} \), \( Q_{\alpha} \) are the volume fraction, number of monomers (i.e., ratio of molecular volume to monomer volume), and normalized partition function for molecular species \( a \), respectively.

General case ( \( M \geq 2 \))

The current version of PSCF allows PS-FTS simulations of some systems with M > 2, subject to a constraint on allowed values of the \( \chi \) parameters: The implementation of the PSA in the current version of PSCF is applicable if and only if the nontrivial eigenvalues of the projected \( \chi \) matrix, \( \lambda_{1}, \ldots , \lambda_{M-1} \), are all negative. When applied a standard AB system, this is equivalent to a requirement that \( \chi > 0 \). In general, when this condition is satisfied, then \( \sigma_{i} = 1 \), and thus

\[ W_{\alpha}({\bf r}) = \omega_{\alpha}({\bf r}) + S_{\alpha} \]

for all \( \alpha = 1, \ldots, M - 1 \), giving real values for all these fields. Because \( W_{\alpha} \) and \( \omega_{\alpha} \) then only differ by a constant shift for these fields, any functional integrals with respect to \( \omega_{\alpha} \) with \( \alpha < M \) can be re-written as an integral with respect to \( W_{\alpha} \) without changing the integral.

To discuss the generalized PSA for such systems, it is convenient to adopt a modified notation analogous to that used for AB systems, in which we use \( W_{-} \) as short-hand for the list of fields

\[ W_{-} = [W_{1}, \ldots, W_{M-1}] \quad, \]

which we refer to collectively as exchange fields, while using \( W_{+} \) as an alternate symbol for the pressure-like field \( W_{M} \). Using this notation, we may express the generalized PSA for such systems as an integral

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

in which \( \int DW_{-} \) denotes a functional integral over the \( M - 1 \) real exchange fields \( W_{-} = [W_1, \ldots, W_{M-1}] \), and in which \( W_{+}^{*} \) denotes a real-valued configuration of \( W_{+}({\bf r}) \) that is chosen to satisfy a partial saddle-point condition requiring that

\[ 0 = \left . \frac{ \delta H } { \delta W_{+}({\bf r}) } \right |_{W_{-}} \]

for all \( {\bf r} \). This partial saddle-point criterion can be shown to be equivalent to a requirement that

\[ \sum_{i=}^{M} \phi_{i}({\bf r}) = 1 \quad. \]

In the case considered here, for which \( \lambda_{1}, \ldots, \lambda_{M-1} \) are all negative, the partial saddle-point approximation yields real values for all average monomer fields \( \phi_{1}({\bf r}), \ldots \phi_{M}({\bf r}) \) and for the Hamiltonian \( H \) in all allowed field configurations.

The partial saddle-point Hamiltonian is given in this more general case is again given by a sum

\[ H = H_{\rm f} + H_{\rm id} \]

in which \( H_{\rm id} = - \ln Z_{\rm id} \) and

\[ H_{\rm f} = \frac{1}{v} \int \! d{\bf r} \; \left \{ \sum_{\alpha=1}^{M-1} \frac{M (W_{\alpha} - S_{\alpha})^2 }{ 2 |\lambda_{\alpha}|} - W_{+}^{*} + \frac{S_{M}}{2} \right \} \quad, \]

where \( \alpha \) is an index for eigenvectors of the projected \( \chi \) matrix, and \( S_{\alpha} \equiv v_{\alpha i}\chi_{ij}e_{j}/M^{2} \) for \( \alpha = 1, \ldots, M \), and \( S_{M} = e_{i}\chi_{ij}e_{j}/M^{2} \).

Status of Algorithms for \( M > 2 \):

The BD and MC algorithms currently provided PSCF can perform simulations of systems with \( M \geq 2 \) if and only if the projected chi matrix has no positive eigenvalues. This statement applies both to the actual BD and MC algorithms used to generate random changes in exchange fields and to the compressor algorithms used to identify partial saddle points. For ABC systems ( \( M = 3 \)), this requirement imposes several constraints on the allowed ranges of values of the chi parameters, as discussed in the article by Morse, Yong and Chen .

Some of the data analysis algorithms provided in the current version of PSCF were, however, designed specifically for systems with \( M = 2 \) and are only usable with such systems. Each "analyzer" class that implements such a restricted algorithm checks whether \( M = 2 \), and will print an error message and then halt execution if the class is used with a system with \( M > 2 \). Currently, it is thus possible to perform PS-FTS calculations on some ABC systems, but some data analysis algorithms cannot be used for this purpose.

Morse, Yong and Chen have proposed a further generalization of the PSA so as to treat cases in which one or more of the \( M - 1 \) nontrival eigenvalues of the projected \( \chi \) matrix are positive. Specificially, they proposed treating the auxiliary fields associated with such eigenvalues in a PSA, in addition to the use of the PSA for the pressure-like field \( W_{M} \). This generalized version of the PSA is not implemented in the current version of PSCF, in which the PSA is currently only applied to \( W_{M} \).

Indeterminancy of pressure in canonical ensemble

In canonical ensemble, in which each molecular species occupies a specified volume fraction, the configuration of the pressure-like field required to satisfy the partial saddle-point (or incompressibility) condition is unique only to within an arbitrary spatially homogeneous constant: In canonical ensemble, if \( W_{+}^{*}({\bf r})\) is a configuration of the pressure-like field that satisfies the partial saddle-point condition for some specified configuration of the other fields, than any other field

\[ W_{+}^{*}({\bf r}) \rightarrow W_{+}^{*}({\bf r}) + C \]

that differs from \( W_{+}^{*}({\bf r}) \) by any spatially homogeneous constant \( C \) can be shown to also satisfy this condition. The resulting indeterminancy also appears in the SCFT equations for a closed, incompressible system, as noted previously.

Different solutions of the partial saddle-point problem that differ by a spatially homogeneous constant correspond to the equivalent physical state that differ only in the value of an irrelevant macroscopic pressure. Such states can be shown yield equal value for the total Hamiltonian \( H \). Such pairs of physically equivalent field configurations do, however, generally yield different values for the components \( H_{\rm id} \) and \( H_{\rm f} \) defined above.

Alternate definitions of Hamiltonian components

To avoid any arbitrariness in the definitions of the ideal and field contributions to \( H \) that could result from non-uniqueness of the pressure field, some classes that compute or analyze these contributions use alternate definitions that do not suffer from this ambiguity. These alternative definitions of the ideal and interaction contributions to \( H \) are denoted here by \( \tilde{H}_{\rm id} \) and \( \tilde{H}_{\rm f} \), and are defined as

\[ \begin{align} \tilde{H}_{\rm id} & = H_{\rm id} - \frac{1}{v}\int W_{+}^{*}({\bf r}) \\ \tilde{H}_{\rm f} & = H_{\rm f} + \frac{1}{v}\int W_{+}^{*}({\bf r}) \end{align} \quad. \]

or, more explicitly,

\[ \begin{align} \tilde{H}_{\rm id} & = \frac{V}{v}\sum_{a} \frac{\overline{\phi}_{a}}{N_{a}} \left [ \ln \left ( \frac{\overline{\phi}_{a}}{Q_{a}} \right ) - 1 \right ] - \frac{1}{v}\int W_{+}^{*}({\bf r}) \\ \tilde{H}_{\rm f} & = \frac{1}{v} \int \! d{\bf r} \; \left \{ \sum_{\alpha=1}^{M-1} \frac{M (W_{\alpha} - S_{\alpha})^2 }{ 2 |\lambda_{\alpha}|} + \frac{S_{M}}{2} \right \} \quad. \end{align} \]

Note that these modified definitions simply move the term proportional to the integral of \( W_{+}^{*}({\bf r}) \) from \( H_{\rm f} \) to \( H_{\rm id} \). This simple modification leaves the sum of these two components unchanged, giving

\[ H = H_{\rm id} + H_{\rm f} = \tilde{H}_{\rm id} + \tilde{H}_{\rm f} \quad. \]

The term proportional to the spatial integral of \( W^{*}({\bf r}) \) in the definition of \( \tilde{H}_{\rm id} \) can be shown to cancel the dependence of the sum of single-molecule free energies on the average pressure, making \( \tilde{H}_{\rm id} \) invariant under any spatially homogeneous shift of \( W^{*}_{+}({\bf r}) \). Note that the resulting definition of \( \tilde{H}_{\rm f} \) has no explicit dependence on \( W_{+}^{*} \), and so is also invariant under any such homogeneous shift in this field. Both \( \tilde{H}_{\rm id} \) and \( \tilde{H}_{\rm int} \) are thus invariant under homogeneous changes in \( W_{+}({\bf r}) \), and so have unique values for each possible configuration of the fluctuating field components \( W_{1}, \ldots, W_{M-1} \). This observation also implies that the total Hamiltonian \( H \) of an incompressible system is invariant under any such homogeneous change in pressure.

For users who read the source code : The quantities \( \tilde{H}_{\rm id} \) and \( \tilde{H}_{\rm f} \) defined above are computed and accessed using member functions of the Pscf::Rp::Simulator class template, and are analyzed by the Pscf::Rp::HamiltonianAnalyzer class template.

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