Path integral formulation

In the path integral formulation the canonical partition function (in one dimension) is written as (Ref. review #4 Eq. 1)

$Q(\beta ,V)=\int {\mathrm {d} }x_{1}\int _{x_{1}}^{x_{1}}Dx(\tau )e^{-S[x(\tau )]}$

where $S[x(\tau )]$ is the Euclidian action, given by (Ref. review #4 Eq. 2)

$S[x(\tau )]=\int _{0}^{\beta \hbar }H(x(\tau ))$

where $x(\tau )$ is the path in time $\tau$ and $H$ is the Hamiltonian. This leads to (Ref. review #4 Eq. 3)

$Q_{P}=\left({\frac {mP}{2\pi \beta \hbar ^{2}}}\right)^{P/2}\int ...\int {\mathrm {d} }x_{1}...{\mathrm {d} }x_{P}e^{-\beta \Phi _{P}(x_{1}...x_{P};\beta )}$

where the Euclidean time is discretised in units of

$\varepsilon ={\frac {\beta \hbar }{P}},P\in {\mathbb {Z} }$

$x_{t}=x(t\beta \hbar /P)$

$x_{P+1}=x_{1}$

and ((Ref. review #4 Eq. 4)

$\Phi _{P}(x_{1}...x_{P};\beta )={\frac {mP}{2\beta ^{2}\hbar ^{2}}}\sum _{t=1}^{P}(x_{t}-x_{t+1})^{2}+{\frac {1}{P}}\sum _{t=1}^{P}V(x_{t})$.

It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical statistical mechanics of polyatomic fluids, in particular flexible ring molecules (Ref. 1).

External links

• Density matrices and path integrals computer code on SMAC-wiki.

General Reading

Reviews

1. R. P. Feynman and A. R. Hibbs, Path-integrals and Quantum Mechanics (McGraw-Hill, New York, 1965) ISBN 0-07-020650-3
2. R. P. Feynman, Statistical Mechanics (Benjamin, Reading, Mass., 1972)
3. David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics 74 pp. 4078-4095 (1981)
4. B. J. Berne and ­D. Thirumalai "On the Simulation of Quantum Systems: Path Integral Methods", Annual Review of Physical Chemistry 37 pp. 401-424 (1986)
5. D. M. Ceperley "Path integrals in the theory of condensed helium", Reviews of Modern Physics 67 279 - 355 (1995)
6. Charusita Chakravarty "Path integral simulations of atomic and molecular systems", International Reviews in Physical Chemistry 16 pp. 421-444 (1997)

Applications

Phase transitions, quantum dynamics, centroids etc.

1. J. R. Melrose and K. Singer "An investigation of supercooled Lennard-Jones argon by quantum mechanical and classical Monte Carlo simulation", Molecular Physics 66 1203-1214 (1989)
2. Jianshu Cao and Gregory A. Voth "The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties", Journal of Chemical Physics 100 pp. 5093-5105 (1994)
3. Jianshu Cao and Gregory A. Voth "Semiclassical approximations to quantum dynamical time correlation functions", Journal of Chemical Physics 104 pp. 273-285 (1996)
4. Rafael Ramírez and Telesforo López-Ciudad "The Schrödinger formulation of the Feynman path centroid density", Journal of Chemical Physics 111 pp. 3339-3348 (1999)
5. C. Chakravarty and R. M. Lynden-Bell "Landau free energy curves for melting of quantum solids", Journal of Chemical Physics 113 pp. 9239-9247 (2000)

References

1. David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics 74 pp. 4078-4095 (1981)