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g.tex
105
g.tex
@ -54,6 +54,7 @@
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\newcommand{\cD}{\mathcal{D}}
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\newcommand{\cE}{\mathcal{E}}
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\newcommand{\cI}{\mathcal{I}}
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\newcommand{\cH}{\mathcal{H}}
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\newcommand{\EPT}{E_{\PT}}
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\newcommand{\laPT}{\lambda_{\PT}}
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@ -534,9 +535,7 @@ See text for additional comments on the time evolution of the path.}
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\label{fig:domains}
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\end{figure}
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Generalizing the single-state case treated previously, let us define the probability of remaining $n$ times in the domain of $\ket{I_0}$ and to exit at $\ket{I} \notin \cD_{I_0}$.
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This probability is given by
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Generalizing the single-state case treated previously, let us define the probability of remaining $n$ times in the domain of $\ket{I_0}$ and to exit at $\ket{I} \notin \cD_{I_0}$
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\be
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\label{eq:eq1C}
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\cP_{I_0 \to I}(n)
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@ -567,7 +566,7 @@ where the operator $F^+_I = P_I T^+ (1-P_I)$, corresponding to the last move co
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Physically, $F$ may be seen as a flux operator through the boundary of $\cD_{I}$.
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Knowing the probability of remaining $n$ times in the domain and, then, to exit to some state, it is possible to obtain
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the probability of being trapped $n$ times in $\cD_{I}$, just by summing over all possible exit states. Thus, we get
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the probability of being trapped $n$ times in $\cD_{I}$, just by summing over all possible exit states:
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\be
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\label{eq:PiN}
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P_{I}(n) = \frac{1}{\PsiG_{I}} \mel{ I }{ \qty(T^+_{I})^{n-1} F^+_{I} }{ \PsiG }.
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@ -602,7 +601,7 @@ Note that it is possible only if the dimension of the domains is not too large (
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In this section we generalize the path-integral expression of the Green's matrix, Eq.~\eqref{eq:G}, to the case where domains are used.
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To do so, we introduce the Green's matrix associated with each domain as follows:
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\be
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G^{(N),\cD}_{ij}= \mel{ i }{ T_i^N }{ j }.
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G^{(N),\cD}_{ij}= \mel{ i }{ \titou{T_i^N} }{ j }.
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\ee
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%Starting from Eq.~\eqref{eq:cn}
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%\be
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@ -628,22 +627,17 @@ It follows that
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\qty[ \prod_{k=0}^{p-1} \mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} } ]
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G^{(n_p-1),\cD}_{I_p i_N},
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\end{multline}
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where $\delta_{i,j}$ is a Kronecker delta. Note that the first contribution, $G^{(N),\cD}_{i_0 i_N}$, corresponds to the case
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$p=0$ and collects all contributions to the Green's matrix coming from paths remaining for ever within the domain of $\ket{i_0}$ (no exit event).
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This contribution is here isolated from the $p$-sum since, as a domain Green's matrix, it will be calculated exactly and will not be suject to a stochastic
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treatment.
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Note also
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that the last state $\ket{i_N}$ is never an exit state because of the very definition of our representation of the paths (if not, it would be
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associated to the next contribution coresponding to $p+1$ exit events).
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where $\delta_{i,j}$ is a Kronecker delta.
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This expression is the path-integral representation of the Green's matrix using only the variables $(\ket{I_k},n_k)$ of the effective dynamics defined over the set of domains.
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The standard formula for $G^{(N)}_{i_0 i_N}$ derived above, Eq.~\eqref{eq:G}, may be considered as the particular case where the
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walker exits of the current state $\ket{i_k}$ at each step (no domains are introduced), leading to a number of
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exit events $p$ equal to $N$. In this case, we have $\ket{I_k}=\ket{i_k}$, $n_k=1$ (for $0 \le k \le N$), and we are left only with the $p$th component of the sum, that is, $G^{(N)}_{i_0 i_N}=\prod_{k=0}^{N-1} \mel{ I_k }{ F_{I_k} }{ I_{k+1} }$, with $F=T$, thus recovering Eq.~\eqref{eq:G}.
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The standard formula for $G^{(N)}_{i_0 i_N}$ derived above [see Eq.~\eqref{eq:G}] may be considered as the particular case where the walker exits of the current state $\ket{i_k}$ at each step (no domains are introduced), leading to a number of exit events $p$ equal to $N$.
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In this case, we have $\ket{I_k}=\ket{i_k}$, $n_k=1$ (for $0 \le k \le N$), and we are left only with the $p$th component of the sum, that is, $G^{(N)}_{i_0 i_N}=\prod_{k=0}^{N-1} \mel{ I_k }{ F_{I_k} }{ I_{k+1} }$, with $F=T$, thus recovering Eq.~\eqref{eq:G}.
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Note that the first contribution $G^{(N),\cD}_{i_0 i_N}$ corresponds to the case $p=0$ and collects all contributions to the Green's matrix coming from paths remaining indefinitely in the domain of $\ket{i_0}$ (no exit event).
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This contribution is isolated from the sum in Eq.~\eqref{eq:Gt} since, as a domain Green's matrix, it is calculated exactly and is not subject to a stochastic treatment.
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Note also that the last state $\ket{i_N}$ is never an exit state because of the very definition of our path representation.
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% (if not, it would be associated to the next contribution corresponding to $p+1$ exit events).
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In order to compute $G^{(N)}_{i_0 i_N}$ by resorting to Monte Carlo techniques, let us reformulate Eq.~\eqref{eq:Gt}
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using the transition probability $\cP_{I \to J}(n)$ introduced above,
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Eq.~\eqref{eq:eq3C}. We first rewrite Eq.~\eqref{eq:Gt} under the form
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In order to compute $G^{(N)}_{i_0 i_N}$ by resorting to Monte Carlo techniques, let us reformulate Eq.~\eqref{eq:Gt} using the transition probability $\cP_{I \to J}(n)$ introduced in Eq.~\eqref{eq:eq3C}.
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We first rewrite Eq.~\eqref{eq:Gt} under the form
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\begin{multline}
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{G}^{(N)}_{i_0 i_N}={G}^{(N),\cD}_{i_0 i_N} + {\PsiG_{i_0}}
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\sum_{p=1}^{N}
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@ -668,40 +662,35 @@ and using the effective transition probability, Eq.~\eqref{eq:eq3C}, we get
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\qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} ) \qty( \prod_{k=0}^{p-1}\cP_{I_k \to I_{k+1}}(n_k) )
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\frac{1}{\PsiG_{I_p}} {G}^{(n_p-1), \cD}_{I_p i_N} },
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\end{multline}
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where, for clarity, $\sum_{{(I,n)}_{p,N}}$ is used as a short-hand notation for the sum,
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$ \sum_{\ket{I_1} \notin \cD_{I_0}, \ldots , \ket{I_p} \notin \cD_{I_{p-1}}}
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\sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}$ with the constraint $\sum_{k=0}^p n_k=N+1$.\\
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\titou{where, for clarity, $\sum_{{(I,n)}_{p,N}}$ is used as a short-hand notation for the sum, $\sum_{\ket{I_1} \notin \cD_{I_0}, \ldots , \ket{I_p} \notin \cD_{I_{p-1}}} \sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}$ with the constraint $\sum_{k=0}^p n_k=N+1$.}
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Under this form ${G}^{(N)}_{i_0 i_N}$ is now amenable to Monte Carlo calculations
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by generating paths using the transition probability matrix, $\cP_{I \to J}(n)$. For example, in the case of the energy, we start from
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Under this form, ${G}^{(N)}_{i_0 i_N}$ is now amenable to Monte Carlo calculations
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by generating paths using the transition probability matrix $\cP_{I \to J}(n)$.
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For example, in the case of the energy, we start from
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\be
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E_0 = \lim_{N \to \infty }
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\frac{ \sum_{i_N} {G}^{(N)}_{i_0 i_N} {(H\PsiT)}_{i_N} }
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{ \sum_{i_N} {G}^{(N)}_{i_0 i_N} {\PsiT}_{i_N} }
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{ \sum_{i_N} {G}^{(N)}_{i_0 i_N} {\PsiT}_{i_N} },
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\ee
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which can be rewritten probabilistically as
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\be
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E_0 = \lim_{N \to \infty }
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\frac{ {G}^{(N),\cD}_{i_0 i_N} + {\PsiG_{i_0}} \sum_{p=1}^{N} \expval{ \qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} ) {\cal H}_{n_p,I_p} }_p}
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{ {G}^{(N),\cD}_{i_0 i_N} + {\PsiG_{i_0}} \sum_{p=1}^{N} \expval{ \qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} ) {\cal S}_{n_p,I_p} }_p}
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{ {G}^{(N),\cD}_{i_0 i_N} + {\PsiG_{i_0}} \sum_{p=1}^{N} \expval{ \qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} ) {\cal S}_{n_p,I_p} }_p},
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\ee
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where $\expval{...}_p$ is the probabilistic average defined over the set of paths $p$ exit events of probability
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$\prod_{k=0}^{p-1} \cP_{I_k \to I_{k+1}}(n_k) $
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and $({\cal H}_{n_p,I_p},{\cal S}_{n_p,I_p})$ two quantities taking into account the contribution of the trial wave function at the end of the path and defined as follows
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\be
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{\cal H}_{n_p,I_p}= \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{(n_p-1),\cD}_{I_p i_N} (H \Psi_T)_{i_N}
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\ee
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and
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\be
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{\cal S}_{n_p,I_p}= \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{(n_p-1), \cD}_{I_p i_N} (\Psi_T)_{i_N}
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\ee
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In practice, the Monte Carlo algorithm is a simple generalization of that used in standard diffusion Monte Carlo calculations.
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Stochastic paths starting at $\ket{I_0}$ are generated using
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the probability $\cP_{I_k \to I_{k+1}}(n_k)$ and are stopped when $\sum_k n_k$ is greater than some target value $N$. Averages of the
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products of weights, $ \prod_{k=0}^{p-1} W_{I_k I_{k+1}} $ times the end-point contributions, ${{(\cal H}/{\cal S})}_{n_p,I_p} $ are computed for each $p$.
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The generation of the paths can be performed using a two-step process. First, the integer $n_k$ is drawn using the probability $P_{I_k}(n)$ [see Eq.~\eqref{eq:PiN}]
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and, then,
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the exit state, $\ket{I_{k+1}}$, is drawn using the conditional probability $\frac{\cP_{I_k \to I_{k+1}}(n_k)}{P_{I_k}(n_k)}$.
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\titou{where $\expval{\cdots}_p$ is the probabilistic average defined over the set of paths $p$ exit events of probability $\prod_{k=0}^{p-1} \cP_{I_k \to I_{k+1}}(n_k)$} and
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\begin{align}
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\cH_{n_p,I_p} & = \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{(n_p-1),\cD}_{I_p i_N} (H \Psi_T)_{i_N},
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\\
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\cS_{n_p,I_p} & = \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{(n_p-1), \cD}_{I_p i_N} (\Psi_T)_{i_N},
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\end{align}
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two quantities taking into account the contribution of the trial wave function at the end of the path.
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\titou{In practice, the Monte Carlo algorithm is a simple generalization of that used in standard diffusion Monte Carlo calculations.}
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Stochastic paths starting at $\ket{I_0}$ are generated using the probability $\cP_{I_k \to I_{k+1}}(n_k)$ and are stopped when $\sum_k n_k$ is greater than some target value $N$.
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Averages of the weight products $ \prod_{k=0}^{p-1} W_{I_k I_{k+1}} $ times the end-point contributions ${(\cH/\cS)}_{n_p,I_p}$ are computed for each $p$.
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The generation of the paths can be performed using a two-step process.
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First, the integer $n_k$ is drawn using the probability $P_{I_k}(n)$ [see Eq.~\eqref{eq:PiN}] and, then, the exit state $\ket{I_{k+1}}$ is drawn using the conditional probability $\cP_{I_k \to I_{k+1}}(n_k)/P_{I_k}(n_k)$.
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%See fig.\ref{scheme1C}.
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%\titou{i) Choose some initial vector $\ket{I_0}$\\
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%ii) Generate a stochastic path by running over $k$ (starting at $k=0$) as follows.\\
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@ -732,7 +721,7 @@ Let us define the energy-dependent Green's matrix
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\be
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G^E_{ij}= \tau \sum_{N=0}^{\infty} \mel{ i }{ T^N }{ j} = \mel{i}{ \qty( H-E \Id )^{-1} }{j}.
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\ee
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The denomimation ``energy-dependent'' is chosen here since
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The denomination ``energy-dependent'' is chosen here since
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this quantity is the discrete version of the Laplace transform of the time-dependent Green's function in a continuous space,
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usually known under this name.\cite{note}
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The remarkable property is that, thanks to the summation over $N$ up to infinity, the constrained multiple sums appearing in Eq.~\eqref{eq:Gt} can be factorized in terms of a product of unconstrained single sums, as follows
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@ -742,7 +731,7 @@ The remarkable property is that, thanks to the summation over $N$ up to infinity
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= \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} \cdots \sum_{n_p=1}^{\infty} F(n_0,\ldots,n_N).
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\end{multline}
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where $F$ is some arbitrary function of the trapping times.
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Using the fact that $G^E_{ij}= \tau \sum_{N=0}^{\infty} G^{(N)}_{ij}$ where $G^{(N)}_{ij}$ is given by Eq.~\eqref{eq:Gt} and summing over the $n_k$-variables, we get
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Using the fact that $G^E_{ij}= \tau \sum_{N=0}^{\infty} G^{(N)}_{ij}$, where $G^{(N)}_{ij}$ is given by Eq.~\eqref{eq:Gt}, and summing over the variables $n_k$, we get
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\begin{multline}
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\label{eq:eqfond}
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{G}^{E}_{i_0 i_N}
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@ -751,7 +740,7 @@ Using the fact that $G^E_{ij}= \tau \sum_{N=0}^{\infty} G^{(N)}_{ij}$ where $G^{
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\qty[ \prod_{k=0}^{p-1} \mel{ I_k }{ {\qty[ P_k \qty( H-E \Id ) P_k ] }^{-1} (-H)(\Id-P_k) }{ I_{k+1} } ]
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{G}^{E,\cD}_{I_p i_N}
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\end{multline}
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where, ${G}^{E,\cD}$ is the energy-dependent domain's Green matrix defined as ${G}^{E,\cD}_{ij} = \tau \sum_{N=0}^{\infty} \mel{ i }{ T^N_i }{ j}$.
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where, ${G}^{E,\cD}$ is the energy-dependent domain's Green matrix defined as ${G}^{E,\cD}_{ij} = \tau \sum_{N=0}^{\infty} \mel{ i }{ \titou{T^N_i} }{ j}$.
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As a didactical example, Appendix \ref{app:A} reports the exact derivation of this formula in the case of a two-state system.
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@ -836,25 +825,24 @@ of the non-linear equation $\cE(E)= E$ in the vicinity of $E_0$.
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In practical Monte Carlo calculations the DMC energy will be obtained by computing a finite number of components $H_p$ and $S_p$ defined as follows
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\be
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\cE^\text{DMC}(E,p_{max})= \frac{ H_0+ \sum_{p=1}^{p_{max}} H^\text{DMC}_p }
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{S_1 +\sum_{p=1}^{p_{max}} S^\text{DMC}_p }
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{S_{\titou{0}} +\sum_{p=1}^{p_{max}} S^\text{DMC}_p }.
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\ee
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For $ p\ge 1$, Eq~\eqref{eq:final_E} gives
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\begin{align}
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H^\text{DMC}_p & = \PsiG_{i_0}\expval{ \qty(\prod_{k=0}^{p-1} W^E_{I_k I_{k+1}}) {\cal H}_{I_p} }
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H^\text{DMC}_p & = \PsiG_{i_0}\expval{ \qty(\prod_{k=0}^{p-1} W^E_{I_k I_{k+1}}) {\cal H}_{I_p} },
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\\
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S^\text{DMC}_p & = \PsiG_{i_0} \expval{ \qty(\prod_{k=0}^{p-1} W^E_{I_k I_{k+1}}) {\cal S}_{I_p} }.
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S^\text{DMC}_p & = \PsiG_{i_0} \expval{ \qty(\prod_{k=0}^{p-1} W^E_{I_k I_{k+1}}) {\cal S}_{I_p} },
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\end{align}
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where
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\be
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{\cal H}_{I_p}= \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{(E,\cD}_{I_p i_N} (H \Psi_T)_{i_N}
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\ee
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and
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\be
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{\cal S}_{I_p}= \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{E, \cD}_{I_p i_N} (\Psi_T)_{i_N}
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\ee
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\begin{align}
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\cH_{I_p} & = \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{E,\cD}_{I_p i_N} (H \Psi_T)_{i_N},
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\\
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\cS_{I_p} & = \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{E, \cD}_{I_p i_N} (\Psi_T)_{i_N}.
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\end{align}
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For $p=0$, the two components are exactly evaluated as
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\begin{align}
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H_0 & = \mel{ I_0 }{ {\qty[ P_{I_0} \qty(H-E \Id) P_{I_0} ]}^{-1} }{ H\PsiT }
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H_0 & = \mel{ I_0 }{ {\qty[ P_{I_0} \qty(H-E \Id) P_{I_0} ]}^{-1} }{ H\PsiT },
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\\
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S_0 & = \mel{ I_0 }{ {\qty[ P_{I_0} \qty(H-E \Id) P_{I_0} ]}^{-1} }{ \PsiT }.
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\end{align}
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@ -1139,8 +1127,7 @@ $\cD(0,1)\cup\cD(1,0)\cup\cD$(2,0) & 36 & $\infty$&1&$-0.75272390$\\
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\end{ruledtabular}
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\end{table}
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As explained above, it is very advantageous to calculate exactly as many $(H_p,S_p)$ as possible in order to avoid the sttistical error on the heaviest
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components.
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As explained above, it is very advantageous to calculate exactly as many $(H_p,S_p)$ as possible in order to avoid the statistical error on the largest components.
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Table \ref{tab2} shows the results both for the case of a single-state main domain and for the domain having the largest average trapping time, namely $\cD(0,1) \cup \cD(1,1)$ (see Table \ref{tab1}).
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Table \ref{tab2} reports the statistical fluctuations of the energy for the simulation of Table \ref{tab1}.
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Results show that it is indeed interesting to compute exactly as many components as possible.
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