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Path Integrals

The path integral formulation, here from the statistical mechanical point of view, is an elegant method by which quantum mechanical contributions can be incorporated within a classical simulation using Feynman path integrals (see the additional reading section). Such simulations are particularly applicable to light atoms and molecules such as hydrogen, helium, neon and argon, as well as quantum rotators such as methane and hydrogen-bonded systems such as water. From a more idealized point of view path integrals are often used to study quantum hard spheres.


In the path integral formulation the canonical partition function (in one dimension) is written as ([1] 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 Euclidean action, given by ([1] Eq. 2) \[S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau)) ~{\mathrm d}\tau\] where \(x(\tau)\) is the path in time \(\tau\) and \(H\) is the Hamiltonian. This leads to ([1] 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 ([1] Eq. 4) \[\Phi_P (x_1...x_P;\beta)= \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P (x_t - x_{t+1})^2 + \frac{1}{P} \sum_{t=1}^P V(x_t)\]

where \(P\) is the Trotter number. In the Trotter limit, where \(P \rightarrow \infty\) these equations become exact. In the case where \(P=1\) these equations revert to a classical simulation. 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[2], due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle \(x_t\) interacts with is neighbours \(x_{t-1}\) and \(x_{t+1}\) via a harmonic spring. The second term provides the internal potential energy.

The following is a schematic for the interaction between atom \(i\) (green) and atom \(j\) (orange). Here we show the atoms having five Trotter slices (\(P=5\)), forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the classical intermolecular pair potential.

1bead classical ij.png

Classical limit (P=1)

5bead pathIntegra ij.png

Path integral (here with P=5)

In three dimensions one has the density operator

\[\hat{\rho} (\beta) = \exp\left[ -\beta \hat{H} \right]\]

which thanks to the Trotter formula we can tease out \(\exp \left[ -\beta (U_{\mathrm {spring}}+ U_{\mathrm{internal}} ) \right]\), where

\[U_{\mathrm {spring}} = \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P | \mathbf{r}_t - \mathbf{r}_{t+1} |^2\]


\[U_{\mathrm{internal}}= \frac{1}{P} \sum_{t=1}^P V(\mathbf{r}_t)\]

The internal energy is given by

\[\langle U \rangle = \frac{3NP}{2\beta}- \langle U_{\mathrm {spring}} \rangle + \langle U_{\mathrm{internal}} \rangle \]

The average kinetic energy is known as the primitive estimator, i.e.

\[\langle K_P \rangle = \frac{3NP}{2\beta}- \langle U_{\mathrm {spring}} \rangle \]

Harmonic oscillator

The density matrix for a harmonic oscillator is given by ([3] Eq. 10-44)

\[\rho(x',x)= \sqrt{ \frac{m \omega}{2 \pi \hbar \sinh \omega \beta \hbar} } \exp \left( - \frac{m \omega}{2 \hbar (\sinh \omega \beta \hbar)^2 } \left( (x^2 + x'^2 ) \cosh \omega \beta \hbar - 2xx'\right)\right)\]

See also refs [4] [5]

Wick rotation and imaginary time

Wick rotation [6]. One can identify the inverse temperature, \(\beta\) with an imaginary time \(it/\hbar\) (see [7] § 2.4).

Rotational degrees of freedom

In the case of systems having (\(d\)) rotational degrees of freedom the Hamiltonian can be written in the form ([8] Eq. 2.1): \[\hat{H} = \hat{T}^{\mathrm {translational}} + \hat{T}^{\mathrm {rotational}}+ \hat{V}\]

where the rotational part of the kinetic energy operator is given by ([8] Eq. 2.2)

\[T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}\]

where \(\hat{L}_i\) are the components of the angular momentum operator, and \(\Theta_{ii}\) are the moments of inertia.

Computer simulation techniques

The following are a number of commonly used computer simulation techniques that make use of the path integral formulation applied to phases of condensed matter


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