For separation of variables in the eigenvalue problem \[\hat{L} \, P(x,y)=\lambda \, P(x,y) \eqno(1)\] the operator \(\hat{L}\) should be represented as a sum of two operators \(\hat{L}_x +\hat{M}_y\), depending only on \(x\) and \(y\) correspondingly [1].¹

Conditions for separation of variables in the Fokker–Planck equations appear to be essentially weaker. For example, in the equation describing the time evolution of the probability distribution \(P\equiv P(x,y)\), \[\frac{\partial P}{\partial t}= \left\{\vphantom{L^2} \hat{L}_{x,y} P \right\}'_x + \left\{ \vphantom{L^2} \hat{M}_{y} P \right\}'_y \,, \eqno(2)\] it is sufficient that the operator \(\hat{M}_{y}\) in the last term depends only on \(y\), while the operator \(\hat{L}_{x,y}\) remains arbitrary. Indeed, setting \(P=P(x) P(y)\) and dividing by \(P(x)\), one has \[-\frac{\partial P(y)}{\partial t} + \left\{ \vphantom{L^2} \hat{M}_{y} P(y) \right\}'_y=\] \[=\frac{P(y)}{P(x)} \frac{\partial P(x)}{\partial t} -\frac{1}{P(x)} \left\{\vphantom{L^2} \hat{L}_{x,y} P \right\}'_x \,. \eqno(3)\] The left-hand side is independent of \(x\), and can be considered as a certain function \(F(y)\). Then \[P(y) \frac{\partial P(x)}{\partial t} -\left\{\vphantom{L^2} \hat{L}_{x,y} P \right\}'_x = F(y) P(x) \eqno(4)\] and integration over \(x\) gives \(F(y)\equiv 0\), since the left-hand side turns to zero, while the integral over \(P(x)\) is equal to unity due to normalization. As a result, the left-hand side and right-hand side of Eq. 3 turn to zero independently, and the equation for \(P(y)\) is separated \[\frac{\partial P(y)}{\partial t} - \left\{ \vphantom{L^2} \hat{M}_{y} P(y) \right\}'_y =0 \,. \eqno(5)\] On the other hand, integrating (3) over \(y\), one has \[\frac{\partial P(x)}{\partial t} - \left\{ \vphantom{L^2} \hat{\cal L}_{x} P(x) \right\}'_x =0 \,, \eqno(6)\] where \[\hat{\cal L}_{x} = \int \hat{L}_{x,y} P(y) dy \,. \eqno(7)\] The given considerations are very general and applicable to any diffusion-type equations. It is essential that in physical applications such equations describe evolution of the probability distribution function, which obeys the corresponding properties (positiveness, decreasing at infinity, normalization to unity). Hence, their right-hand side is always a sum of full derivatives, in order to provide the conservation of probability (e.g. integration of Eq.2 over \(x\) and \(y\) gives zero in both sides of the equation) ². As a result, the conditions for separation of variables appear to be always weaker than for equation (1). In our opinion, this fact should be mentioned in any courses of the mathematical physics; unfortunately, it is not the case.

The separation of variables in the Fokker–Planck equations was discussed in the comparatively new papers (e.g. [4]–[6]), but under rather restricted assumptions. The equation of type (2) arises in the theory of 1D localization, where it describes the evolution of the mutual distribution \(P(\rho,\psi)\) of the Landauer resistace \(\rho\) and the phase variable \(\psi=\theta-\varphi\), where \(\theta\) and \(\varphi\) are phases entering the transfer matrix (see Eq.28 in [7] and the comments after it). Analogous situation is expected in description of quasi-1D systems in the framework of the generalized version [8] of the Dorokhov–Mello–Pereyra-Kumar equation [9], [10]. It looks probable that analogous equations arose in other fields of physics and in some cases the fact of separation of variables was revealed by corresponding authors. However, the general character of this result was not emphasized, and it remains unknown to a wide audience.

It should be noted that there is essential difference in the physical and mathematical views on the matter in question. In mathematical literature, the evolution equations \(\partial P/\partial t = \hat{L} P\) are usually considered under assumption that the operator \(\hat{L}\) in the right-hand side is an arbitrary differential operator of the elliptic type ³. From this point of view, equation (2) looks somewhat artificially, as a partial case with doubtful actuality. However, the Fokker-Planck equations in physical applications are written not for an abstract function \(P\), but for the probability distribution. In the latter case, representation of the right-hand side in the form of a sum of full derivatives corresponds to the most general situation. Appropriately, the presented here result has the perfectly general character and refers to arbitrary diffusion type equations. A separation of variables in the physical problem is not simply a technical trick, but a fact with serious consequences: e.g. the separated equation for \(P(\psi)\) in the above example provides the existence of the stationary distribution for the phase variable \(\psi\), which is only essential for the given problem.

The author is idebted to V.V.Losyakov for a valuable remark leading to Footnote 1.

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