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Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of equations:

$\frac{\partial x_n}{\partial t} = -\frac{\partial}{\partial x_n} U(\vec{x}) + \sqrt{2c} \eta_n$

the problem can be reformulated in terms of the probability distribution $P(\vec{x},t)$, through the following fokker-planck equation:

$\frac{\partial P(\vec{x},t)}{\partial t} = \bigg( - \sum_{i=1}^N \frac{\partial}{\partial x_i} \big( \frac{\partial}{\partial x_i}U(\vec{x})\big) + c \sum_{i,j=1}^N \frac{\partial^2}{\partial x_i \partial x_j} \bigg) P(\vec{x},t)$

The equation above admits the following stationary solution:

$P^s(\vec{x}) = \mathcal{N} e^{\frac{-U(\vec{x})}{c}}$

Is there a simple way to convince yourself that, in this case, given any initial distribution I always converge only to above $P^s(\vec{x})$?

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of equations:

$\frac{\partial x_n}{\partial t} = -\frac{\partial}{\partial x_n} U(\vec{x}) + \sqrt{2c} \eta_n$

the problem can be reformulated in terms of the probability distribution $P(\vec{x},t)$, through the following fokker-planck equation:

$\frac{\partial P(\vec{x},t)}{\partial t} = \bigg( \sum_{i=1}^N \frac{\partial}{\partial x_i} \big( \frac{\partial}{\partial x_i}U(\vec{x})\big) + c \sum_{i,j=1}^N \frac{\partial^2}{\partial x_i \partial x_j} \bigg) P(\vec{x},t)$

The equation above admits the following stationary solution:

$P^s(\vec{x}) = \mathcal{N} e^{\frac{-U(\vec{x})}{c}}$

Is there a simple way to convince yourself that, in this case, given any initial distribution I always converge only to above $P^s(\vec{x})$?

consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of equations:

$\frac{\partial x_n}{\partial t} = \frac{\partial}{\partial x_n} U(\vec{x}) + \sqrt{2c} \eta_n$

the problem can be reformulated in terms of the probability distribution $P(\vec{x},t)$, through the following fokker-planck equation:

$\frac{\partial P(\vec{x},t)}{\partial t} = \bigg( - \sum_{i=1}^N \frac{\partial}{\partial x_i} \big( \frac{\partial}{\partial x_i}U(\vec{x})\big) + c \sum_{i,j=1}^N \frac{\partial^2}{\partial x_i \partial x_j} \bigg) P(\vec{x},t)$

The equation above admits the following stationary solution:

$P^s(\vec{x}) = \mathcal{N} e^{\frac{-U(\vec{x})}{c}}$

Is there a simple way to convince yourself that, in this case, given any initial distribution I always converge only to above $P^s(\vec{x})$?

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of equations:

$\frac{\partial x_n}{\partial t} = -\frac{\partial}{\partial x_n} U(\vec{x}) + \sqrt{2c} \eta_n$

the problem can be reformulated in terms of the probability distribution $P(\vec{x},t)$, through the following fokker-planck equation:

$\frac{\partial P(\vec{x},t)}{\partial t} = \bigg( - \sum_{i=1}^N \frac{\partial}{\partial x_i} \big( \frac{\partial}{\partial x_i}U(\vec{x})\big) + c \sum_{i,j=1}^N \frac{\partial^2}{\partial x_i \partial x_j} \bigg) P(\vec{x},t)$

The equation above admits the following stationary solution:

$P^s(\vec{x}) = \mathcal{N} e^{\frac{-U(\vec{x})}{c}}$

Is there a simple way to convince yourself that, in this case, given any initial distribution I always converge only to above $P^s(\vec{x})$?

consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of equations:

$\frac{\partial x_n}{\partial t} = \frac{\partial}{\partial x_n} U(\vec{x}) + \sqrt{2c} \eta_n$

the problem can be reformulated in terms of the probability distribution $P(\vec{x},t)$, through the following fokker-planck equation:

$\frac{\partial P(\vec{x},t)}{\partial t} = \bigg( - \sum_{i=1}^N \frac{\partial^2}{\partial x_i^2} U(\vec{x}) + c \sum_{i,j=1}^N \frac{\partial^2}{\partial x_i \partial x_j} \bigg) P(\vec{x},t)$

The equation above admits the following stationary solution:

$P^s(\vec{x}) = \mathcal{N} e^{\frac{-U(\vec{x})}{c}}$

Is there a simple way to convince yourself that, in this case, given any initial distribution I always converge only to above $P^s(\vec{x})$?

consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of equations:

$\frac{\partial x_n}{\partial t} = \frac{\partial}{\partial x_n} U(\vec{x}) + \sqrt{2c} \eta_n$

the problem can be reformulated in terms of the probability distribution $P(\vec{x},t)$, through the following fokker-planck equation:

$\frac{\partial P(\vec{x},t)}{\partial t} = \bigg( - \sum_{i=1}^N \frac{\partial}{\partial x_i} \big( \frac{\partial}{\partial x_i}U(\vec{x})\big) + c \sum_{i,j=1}^N \frac{\partial^2}{\partial x_i \partial x_j} \bigg) P(\vec{x},t)$

The equation above admits the following stationary solution:

$P^s(\vec{x}) = \mathcal{N} e^{\frac{-U(\vec{x})}{c}}$

Is there a simple way to convince yourself that, in this case, given any initial distribution I always converge only to above $P^s(\vec{x})$?