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This assumption is needed in the proof of the theorem of S-TDCDFT (see Ref.).
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In Eq. 35, ∇j contains the derivatives with respect to the coordinates of the jth particle, i.e., in three dimensions ∇j = (∂ xj, ∂ yj, ∂ zj).
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In Eq. 35, ∇j contains the derivatives with respect to the coordinates of the jth particle, i.e., in three dimensions ∇j = (∂ xj, ∂ yj, ∂ zj).
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Here two kinds of correlations are present. The "usual" correlations introduced by the particles interaction and statistics and the "statistical" correlation introduced in the system by the coupling with the environment. Needless to say these correlations are usually entangled, so that any approximation used to reduce the first to an external contribution will probably neglect important contributions from the second and vice versa. On the other hand, we do not know of any approximation to Ah-xc that contains both correlations, and further investigation is thus necessary.
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Here two kinds of correlations are present. The "usual" correlations introduced by the particles interaction and statistics and the "statistical" correlation introduced in the system by the coupling with the environment. Needless to say these correlations are usually entangled, so that any approximation used to reduce the first to an external contribution will probably neglect important contributions from the second and vice versa. On the other hand, we do not know of any approximation to Ah-xc that contains both correlations, and further investigation is thus necessary.
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Here, remember that dxx- x-dx dx/2x- (dx) 2 /8 x3 + which follows from the standard Taylor expansion of the function x for x 0.
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Here, remember that dxx- x-dx dx/2x- (dx) 2 /8 x3 + which follows from the standard Taylor expansion of the function x for x 0.
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35
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54449087265
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A density-matrix formalism would be even more computationally demanding, requiring the solution of (CNM +2) × (CNM -1) /2 coupled differential equations, even after taking into account the constraints of Hermiticity and unit trace of the density matrix.
-
A density-matrix formalism would be even more computationally demanding, requiring the solution of (CNM +2) × (CNM -1) /2 coupled differential equations, even after taking into account the constraints of Hermiticity and unit trace of the density matrix.
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36
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54449086153
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Note that the theorem of S-TDCDFT is still valid, and Eq. 57 would be exact (and not an approximation), if we choose the bath operators to act on single-particle states (or the density) to begin with.
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Note that the theorem of S-TDCDFT is still valid, and Eq. 57 would be exact (and not an approximation), if we choose the bath operators to act on single-particle states (or the density) to begin with.
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In calculating the time evolution with the SSE we make use of the techniques discussed in Sec. 5.
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In calculating the time evolution with the SSE we make use of the techniques discussed in Sec. 5.
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We expect that, for noninteracting particles, the deviation between the dynamics obtained via the density-matrix equation and the SSE scales as 1/m if m is the number of independent runs on which we average the SSE.
-
We expect that, for noninteracting particles, the deviation between the dynamics obtained via the density-matrix equation and the SSE scales as 1/m if m is the number of independent runs on which we average the SSE.
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The initial state is pure and the bath is selecting only a particular state thus forcing the system toward another pure state. Moreover, we can prove that if the system evolves from the ground state, the stochastic part vanishes on this state and then the boson gas remains in the ground state of the interacting Hamiltonian.
-
The initial state is pure and the bath is selecting only a particular state thus forcing the system toward another pure state. Moreover, we can prove that if the system evolves from the ground state, the stochastic part vanishes on this state and then the boson gas remains in the ground state of the interacting Hamiltonian.
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Only the dynamics obtained from the SSE is reported in Fig. 6.
-
Only the dynamics obtained from the SSE is reported in Fig. 6.
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