γ
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GPU based Monte-Carlo simulation of the 4 Path Integrals for free
particles (n = 1024). it is the number of MC iterations per frame.
The thermostat algorithm and linear congruential random number generator are
used. To simplify visualization paths are clamped to (0,1) interval.
Really it is just closed random walks on torus.

draft

Path Integral for free particles

W = Z^{ -1} e^{-βE},
β = 1/T,
E = ∑_{ i} (x_{i+1} + x_{i
})^{2}/n. In one MC pass we concider only even or odd (independent) beads.
Let for x_{i} bead
x_{o} = (x_{i+1} + x_{i-1 })/2,
Δ = (x_{i+1} - x_{i-1 })/2,
x_{i+1} = x_{o} + Δ,
x_{i-1} = x_{o} - Δ. Then
ε = (x_{i} - x_{i+1 })^{2}
+ (x_{i} - x_{i-1 })^{2} =
(x_{i} - x_{o} + Δ)^{2} + (x_{i} -
x_{o} - Δ)^{2} =
2(δ^{2} + Δ^{2}),
where δ = x_{i} - x_{o} and
w ~ e^{-2βδ2}.
We use random perturbation in every bead
δ = γ(∑_{ k}^{6}
R_{k} - 3) to generate random path.
WebCL Demosupdated 30 Oct 2011