Introduction:
Langevin equation is a stochastic differential equation. It is named after the French physicist Paul Langevin, who developed it in 1908. The equation is used to describe the motion of a particle in a viscous medium when subjected to a random force. It is widely used in the study of physical and chemical systems, such as Brownian motion, polymer dynamics, and protein folding. In this article, we will discuss the Langevin equation, its derivation, and some of its applications in physical chemistry.
The Langevin Equation:
The Langevin equation is a stochastic differential equation that describes the motion of a particle in a viscous medium when subjected to a random force. The equation is given by:
m dv/dt = – γ v + F(t)
where m is the mass of the particle, v is its velocity, γ is the friction coefficient, and F(t) is the random force. The random force is assumed to be a Gaussian white noise with zero mean and a correlation function given by:
= 2 kT γ δ(t – t’)
where denotes the ensemble average, k is the Boltzmann constant, T is the temperature, and δ(t – t’) is the Dirac delta function.
Derivation of Langevin Equation:
The Langevin equation can be derived from the Newtonian equation of motion for a particle in a viscous medium. The Newtonian equation of motion is given by:
m dv/dt = – γ v + F
where F is the external force acting on the particle. In the case of a Brownian particle, F is the random force due to the collisions with the molecules of the medium.
The random force can be modeled as a Gaussian white noise with zero mean and a correlation function given by:
= 2 kT γ δ(t – t’)
The Langevin equation can be obtained by averaging the Newtonian equation of motion over the random force:
= – γ +
The ensemble average of the random force is zero, so = 0. The ensemble average of the product of the random force with the velocity is given by:
= 0
since the random force is uncorrelated with the velocity. Therefore, the averaged equation of motion becomes:
= – γ
which can be rearranged as:
m dv/dt + γ v = 0
This equation can be solved to obtain the velocity as a function of time:
v(t) = v(0) exp(-γt/m)
where v(0) is the initial velocity. The Langevin equation is obtained by adding the random force to the Newtonian equation of motion:
m dv/dt = – γ v + F(t)
Applications of Langevin Equation:
The Langevin equation has many applications in physical chemistry. Some of the important applications are discussed below:
Brownian motion: The Langevin equation is used to describe the Brownian motion of particles in a fluid. Brownian motion is the random motion of particles due to the collisions with the molecules of the fluid. The Langevin equation provides a quantitative description of Brownian motion and is widely used in the study of diffusion, viscosity, and thermal fluctuations.
Polymer dynamics: The Langevin equation is used to describe the dynamics of polymer chains in a solvent. Polymer chains are flexible and can adopt many different conformations. The Langevin equation provides a framework to study the dynamics of polymers and their behavior in solution.
Protein folding: The Langevin equation is used to study the folding of proteins. Protein folding is a complex process that involves many intermediate states. The Langevin equation provides a quantitative description of the forces that drive protein folding and the kinetics of the process.
Colloidal particles: The Langevin equation is used to study the motion of colloidal particles in a fluid. Colloidal particles are small particles that are suspended in a fluid. The Langevin equation provides a framework to study the motion of colloidal particles and their interaction with the surrounding fluid.
Conclusion:
The Langevin equation is a powerful tool in physical chemistry. It provides a quantitative description of the motion of particles in a viscous medium when subjected to a random force. The equation has many applications in the study of Brownian motion, polymer dynamics, protein folding, and colloidal particles. The Langevin equation has been used to make many important discoveries in physical chemistry, and it continues to be an important tool for researchers in this field.