Variational supersymmetric approach and Gram–Schmidt process for evaluating Fokker–Planck probabilities

In this work, an alternative method for solving eigenvalue equations is investigated, with a specific application to the Schrödinger-type Fokker–Planck equation. This method is based on combined eigenfunctions through the Gram–Schmidt orthogonalization process, coupled with the well-formalized factorization technique in supersymmetric quantum mechanics. Eigenvalues are obtained via the variational method, using numerical computation. The aim is to obtain solutions for two polynomial potentials of the form $V_1\left(x\right)=\frac{x^6}{6}-\frac{x^4}{4}$ and $V_2\left(x\right)=\frac{x^4}{4}-\frac{x^3}{5}-\frac{x^2}{2}$, in order to obtain the probability distributions at different times $t$ and initial conditions represented by $x_0$. The results for the symmetric potential $V_1\left(x\right)$ are compared with values found in the literature. For the asymmetric potential $V_2\left(x\right)$, the solution is compared only with numerical results, also demonstrating a low margin of error. In both cases, the proposed technique generates probability distributions that respect the typical behavior of the Fokker–Planck equation, with percentage errors below 0.5% compared to reference methods.

The study demonstrates that the approach based on Gram–Schmidt orthogonalization and the variational method is an effective and alternative tool for solving the Fokker–Planck equation in systems described by polynomial potentials, reliably reproducing results for both ground and excited states.

As a possible application, the method is employed to investigate the folding dynamics of the protein $CI2$, treating folding as a diffusive process governed by the Fokker–Planck equation. A bistable polynomial potential of the form $V\left(x\right)=a_0{x}^4+b_0{x}^3+c_0{x}^2+d_{0}x+g_0$, obtained from computational simulations of the thermodynamic free energy curve, is employed to model the energy profile of the protein. This methodology enables the analysis of the time evolution of the probability distribution under different initial conditions.