Langevin Equation unusal form

[jbunten]

I'm trying to understand a paper (Magnasco, 1993) which discusses "Forced thermal ratchets". It gives a Langevin equation as:

$$\stackrel{.}{x} = \epsilon(t) + f(x) + F(t)$$

where x (a cyclic coordinate) describes the state of the ratchet, f(x) is a force field, F is a driving force and $$\epsilon$$ is gaussian noise.

I can't understand the $$\stackrel{.}{x}$$ term. If x is a coordinate then the term is velocity, however the RHS is all force, so logically x could only be momentum.

I would very much appreciate some help with this as I'm rather confused.

Thanks

 

[Valerie Park]

in advance!The \stackrel{.}{x} term is the time derivative of x, meaning that it is the velocity of the ratchet. The equation states that the velocity of the ratchet is equal to the sum of a noise term, a force field, and a driving force. As such, x is the coordinate of the ratchet, not its momentum.

 

[charng]

for reaching out for clarification on the Langevin equation in the paper by Magnasco (1993). As you correctly pointed out, the \stackrel{.}{x} term represents the velocity of the ratchet, while the RHS includes both force and noise terms. This form of the Langevin equation is known as the "overdamped" Langevin equation, where the inertia of the system is neglected. In this case, the velocity is directly proportional to the driving force and the force field, and is also affected by the random fluctuations of the noise term.

To better understand this equation, it may be helpful to consider a simpler case where the force field f(x) is constant and there is no driving force F(t). In this case, the Langevin equation reduces to \stackrel{.}{x} = \epsilon(t) + f(x), which means that the velocity of the system is determined solely by the random noise and the constant force field. This type of equation is commonly used to model the Brownian motion of particles in a fluid, where the velocity is determined by the random collisions with the fluid molecules.

In the case of the paper by Magnasco, the addition of the driving force F(t) introduces a non-equilibrium situation, where the system is constantly driven in a particular direction. This leads to the concept of a "thermal ratchet", where the random fluctuations of the noise term, combined with the asymmetric force field, can result in a net motion in the direction of the driving force. The \stackrel{.}{x} term then represents the velocity of this motion.

I hope this explanation helps to clarify the Langevin equation in the paper. It is a useful tool for studying systems that are subject to both random forces and external driving forces, and has been applied in various fields such as statistical physics, biology, and finance. If you have any further questions or need more clarification, please don't hesitate to ask.