Let $$P$$ be a positive scalar function and $$\mathbf{v}(\mathbf{x})$$ is an assigned smooth vector field. The quantity $$P(t,\mathbf{x})$$ evolves according to a transport equation of the kind $$\partial_t P(\mathbf{x},t) = -\nabla \cdot [ \mathbf{v}(\mathbf{x}) P(\mathbf{x},t) - \nabla P(\mathbf{x},t) ]$$ The steady state solution $$P(\mathbf{x})$$ is given by
$$\nabla \cdot [ \mathbf{v}(\mathbf{x}) P(\mathbf{x}) - \nabla P(\mathbf{x}) ] = 0$$ If $$\mathbf{v} = -\nabla U$$, then the formal solution has the usual Gibbs form $$P \propto e^{-U }$$ Question: assume that the above equation is defined on the 2D (flat) torus $$\mathbb{T}^2$$. How to deal with the case in which $$\mathbf{v} = -\nabla U + \mathbf{q}$$, where $$U$$ has the periodicity imposed by $$\mathbb{T}^2$$ and $$\mathbf{q} =(q_x,q_y)$$ is a constant vector field?

More precisely, I'd like to find the class of solutions of $$\nabla \cdot [ \mathbf{q} P(x,y) - P(x,y) \nabla U(x,y) - \nabla P(x,y) ] = 0$$ with the periodic constraints typical of the flat torus $$[0,1]\times [0,1]$$, i.e.

$$P(0,y) = P(1,y)$$, $$\quad P(x,0) = P(x,1)$$, $$\qquad U(0,y) = U(1,y)$$, $$\quad U(x,0) = U(x,1) \,$$.

If $$\mathbf{q} = 0$$, then the solution $$P \propto e^{-U }$$ works. My feeling is that the problem with $$\mathbf{q} \neq 0$$ is not trivial because of the topology of $$\mathbb{T}^2$$: the constant field $$\mathbf{q}$$ is clearly periodic (so it can live on the torus) but has no periodic potential (i.e. the potential should be $$-q_x x -q_y y$$ that is not periodic).

EDIT: I found this question about divergence-free fields on a torus. In fact, the field $$\mathbf{q} P(x,y) - P(x,y) \nabla U(x,y) - \nabla P(x,y)$$ is required to be divergence-free. Also these notes are interesting and deal with the diffusion on the flat torus (pag. 80).

In terms of differential forms the problem should be (correct me, I am not an expert):

$$J = q P -P dU - d P \qquad \qquad div(J)=0$$

where $$J$$ is a current 1-form, $$q$$ is a constant 1-form, $$P$$ and $$U$$ are 0-forms. In particular, we should have that $$J = R dg$$, where $$g$$ is a 0-form and $$R$$ is a 90-degrees rotation (i.e. it is the Hodge dual operator in two dimensions, $$J = *dg$$). Equivalently, by using a somewhat improper 3D terminology, $$J = rot(A)$$, where $$A=(0,0,g)$$. Maybe the language of differential forms helps understanding why it is not so easy to find a solution when the $$\mathbf{q}$$ term is switched on.

Since this problem is related to systems studied in Physics (transport equations, Fokker-Planck equations), I posted a related question on physics SE.

• Related question about steady state of the diffusion equation: math.stackexchange.com/q/3723797/532409 Commented Jul 16, 2020 at 21:45
• The letter $d$ is a horrid choice for a constant when we're doing calculus. So now turning a "locally exact" solution into a "globally exact" solution is dependent on the topology of the torus. You need to know that the line integrals of the appropriate dual vector field vanish over both circles. Commented Jul 16, 2020 at 22:09
• Sorry, I will edit with some better constant. Sometimes that coefficient is called $d$ because it is the diffusion coefficient. How is this dual vector field defined? Commented Jul 16, 2020 at 22:31
• We need the divergence $1$-form, not the curl $1$-form, so you have to rotate the vector field $\pi/2$ to turn a flux integral into a work integral. Commented Jul 16, 2020 at 22:56
• Related: math.stackexchange.com/q/4182338/532409 "Heat Kernel on a compact manifold without boundary". Important remark: the torus is "stochastically complete": stochastic completeness is the property for a stochastic process to have "infinite life-time" (the total probability of the particle being found in the state space is constantly equal to 1, i.e. particles do not decay). Interesting and related paper: "Langevin diffusions on the torus: estimation and applications" arxiv.org/abs/1705.00296 Commented Apr 20, 2022 at 14:37