These codes implement Feynman and Klenert's 1986 PRA "Effective classical partition function", as a set of Julia codes which can operate on arbitrary 1D potentials. Variational optimisation is provided by automatic differentiation (forward mode) of the 'auxillary potential' Wtilde.
This method 'approximates the classical potential from below', by a simple Gaussian smearing procedure applied to the bare (classical) potential. The method is variational.
The full quantum problem is solved by path integration. The exact integrals cannot be solved, but an approximate harmonic integral can be solved exactly. As with Feynman's polaron solution, a variational connection is made, allowing one freedom to vary the parameters of the approximate harmonic system to minimise a free energy. Having done this, you have an 'effective classical potential' (W) which can be integrated over to get the partition function, etc.
This method contains most of the quantum-fuzziness, but misses out details of the state symmetry and tunnelling behaviour. This is most notable in the errors for the ground state.
In Gribbin's biography (see [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=feynman]), it is mentioned mentioned that they used a Sinclair ZX Spectrum to run the codes on. (The same 8-bit 1980s home-computer I learnt to program with!) Using Julia & automatic-differentiation on this problem is perhaps slightly over the top - but as the methods are available and seem robust and more than fast enough, they will do!
But at least we can plot the figures in colour these days.
My motivation to do this is:
- because it's cool
- this might be useful
- initially for calculating tunnelling and delocalisation of the nuclear wavefunction in double wells (dynamic stabilisation of soft modes in perovskites)
- the idea of smearing out the potential, is exactly what I've been doing for semi-classical models of recombination
- maybe it could be extended to n-dimensional problems, such as a full set of anharmonic phonons
- I believe it forms the core of the 'centroid' approximation in Path Integral MC, which I do not understand, but would like to.
Central paper is [https://doi.org/10.1103/PhysRevA.34.5080 ] Effective classical partition functions. R. P. Feynman and H. Kleinert. Phys. Rev. A 34, 5080. Published 1 December 1986
This introduces the method, which grows out of the variational approach described in referenced 1 therein, which is the Path Integral book by Feynman and Hibbs (section 10.3, as referenced within the text). Here they apply the method to two test cases of an anharmonic potential, and a double well potential. They also mention applying it to singular potentials, briefly give a Green's Function for response to an external perturbation, and show how it can be generalised to higher dimensions.
Kleinert followed this up with two publications in the early 1990s, which added higher-order diagrams and significantly reduced the error for the ground state. Kleinert describes the second paper in his book as 'a systematic and uniformaly convergent variational perturbation expansion'.
H. Kleinert Improving the Variational Approach to Path Integrals Phys. Lett. B 280, 251 (1992)
and
H. Kleinert Systematic Corrections to Variational Calculation of Effective Classical Potential Phys. Lett. A 173, 332 (1993)
H. Kleinert Path Integratls 5th Edition (2009) Chapter 5 - Variational Perturbation Theory
[https://doi.org/10.1103/PhysRevA.34.5080 ] Effective classical partition functions. R. P. Feynman and H. Kleinert. Phys. Rev. A 34, 5080.
The form for the double well potential (page 34, RHS, third paragraph starting 'Another example is the double-well...'), should read
V(x)=-\frac{1}{2} x^2 + \frac{1}{4} g x^4 + \frac{1}{4g}.
Strangely this is correct in the captions of figure 2 and 3!