From Langevin to Fokker-Planck

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In summary, the conversation discusses the need for a derivation of the Langevin equation for a Brownian particle, which is equivalent to the Fokker-Planck equation for the phase-space distribution function. The person is seeking a derivation that includes a Langevin equation with second derivatives and an outer potential. They also mention a book, "Handbook of Stochastic Methods" by C. Gardiner, as a good resource for understanding these concepts. They offer to help with the derivation if needed.
  • #1
Jezuz
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Hi.
I'm studying quantum Brownian motion right now and I need to see that the (classical) Langevin equation for a Brownian particle is equivalent to the Fokker-Planck equation for the phase-space distribution function of the same particle.
Does anyone know where I can find such a derivation? I've been looking all over the internet for it but usually they start with a Langevin equation containging only first derivatives (that is the have excluded the possible outer potential felt by the particle).
I need the derivation for the case where i have a Langevin equation of the type:
m \ddot x(t) + \gamma \dot x(t) + V(x(t)) = F(t)
(written in LaTeX syntax).
I would be very grateful for help!
Alternatively, since I have the derivation for the Langevin equation starting with a Lagrangian for a particle interacting linearly with a bath of harmonic oscillators (initially in thermal equilibrium), I could also accept a derviation of the Fokker-Planck equation starting with the same assumptions.
 
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  • #2
For such stochastic methods, a very good book is "Handbook of Stochastic Methods" by C. Gardiner. Written by an expert in the field, this book is very physical and to the point unlike many other excessively mathematical treatments of stochastic processes.

If you're interested, I can step you through the derivation here, but you should get that book for sure.
 
  • #3
Okej!
Thank you very much. I'll have a look at that book. Found it at the university library. If there is something I get stuck with I might ask you again :)
 

1. What is the Langevin equation?

The Langevin equation is a stochastic differential equation that describes the movement of a particle in a fluid under the influence of random forces. It was first proposed by Paul Langevin in 1908 and is commonly used in physics, chemistry, and biology to model various phenomena.

2. What is the Fokker-Planck equation?

The Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic process. It was independently derived by Adriaan Fokker and Max Planck in 1914 and is widely used in statistical mechanics and other fields to study the behavior of complex systems.

3. How are the Langevin and Fokker-Planck equations related?

The Langevin equation and the Fokker-Planck equation are intimately connected. The Langevin equation is a stochastic differential equation that describes the dynamics of a system, while the Fokker-Planck equation describes the probability distribution of the system at different points in time. The Fokker-Planck equation is derived from the Langevin equation and provides a more comprehensive understanding of the system's behavior.

4. What is the significance of the Langevin to Fokker-Planck transformation?

The Langevin to Fokker-Planck transformation is a mathematical technique used to transform the Langevin equation into the Fokker-Planck equation. This transformation is important because it allows for a more detailed analysis and understanding of the system's behavior. It also makes it possible to solve the Fokker-Planck equation using various mathematical methods.

5. How are the Langevin and Fokker-Planck equations used in practical applications?

The Langevin and Fokker-Planck equations have a wide range of applications in various fields, including physics, chemistry, biology, and finance. They are used to model and analyze complex systems and phenomena, such as Brownian motion, molecular diffusion, and stock market fluctuations. They are also used in engineering to design and optimize systems and processes.

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