Path Integral Monte Carlo

Feynman paths · imaginary time · quantum-classical isomorphism

Feynman Paths (x vs τ)

Position Density ρ(x)

Potential V(x)



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⟨E⟩ thermal
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⟨x²⟩
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Accept rate
PIMC: Feynman's path integral maps quantum partition function Z = Tr[e^{-βH}] onto a classical ring polymer with P beads (imaginary time slices). Each bead interacts harmonically with neighbors (kinetic term) and with the potential V(x). Monte Carlo samples the Boltzmann distribution of ring polymers. S_E = Σ[mP/2β²·(x_k - x_{k+1})² + β/P·V(x_k)]