Glossary

12.4 Brownian Motion and Langevin Equation

3 min read•july 22, 2024

describes the random movement of particles in fluids, caused by collisions with molecules. This phenomenon, named after botanist , was explained by Einstein and confirmed by , providing evidence for the atomic nature of matter.

The models Brownian motion using Newton's second law, incorporating and random forces. Solutions to this equation yield insights into particle behavior, such as and velocity autocorrelation, which are crucial in understanding and .

Brownian Motion

Characteristics of Brownian motion

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  • Exhibits erratic, random motion of particles suspended in a fluid
    • Results from collisions with the surrounding fluid molecules (water, air)
  • Follows a continuous, but non-differentiable path
    • Particle trajectory is jagged and irregular
  • Coined after botanist Robert Brown, who observed the motion of pollen grains in water (1827)
  • Einstein's explanation (1905) provided evidence for the atomic nature of matter
    • Reconciled thermodynamics with the of gases
  • Perrin's experiments (1908) confirmed Einstein's predictions
    • Measured the mean-square displacement of particles
    • Awarded the Nobel Prize in Physics (1926) for his work

Derivation of Langevin equation

  • Applies Newton's second law for a particle of mass mm: mdvdt=F(t)m \frac{dv}{dt} = F(t)
  • Stochastic force F(t)F(t) consists of two components:
    • Viscous drag: −γv(t)-\gamma v(t), where γ\gamma represents the drag coefficient
      • Opposes the particle's motion through the fluid
    • : η(t)\eta(t), representing collisions with fluid molecules
      • Fluctuates rapidly and averages to zero
  • Langevin equation: mdvdt=−γv(t)+η(t)m \frac{dv}{dt} = -\gamma v(t) + \eta(t)
    • Governs the evolution of the particle's velocity v(t)v(t) over time
  • Random force η(t)\eta(t) has the following properties:
    • Zero mean: ⟨η(t)⟩=0\langle \eta(t) \rangle = 0
      • Averages out to zero over long timescales
    • Delta-correlated: ⟨η(t)η(t′)⟩=2γkBTδ(t−t′)\langle \eta(t) \eta(t') \rangle = 2 \gamma k_B T \delta(t-t')
      • kBk_B: Boltzmann constant, TT: temperature
      • Collisions at different times are uncorrelated

Solutions for Langevin equation

  • Mean-square displacement (MSD): ⟨Δx2(t)⟩=⟨[x(t)−x(0)]2⟩\langle \Delta x^2(t) \rangle = \langle [x(t) - x(0)]^2 \rangle
    • Quantifies the average squared distance traveled by the particle
    • Solution: ⟨Δx2(t)⟩=2kBTγ(t−mγ(1−e−γt/m))\langle \Delta x^2(t) \rangle = \frac{2k_BT}{\gamma} \left( t - \frac{m}{\gamma} (1 - e^{-\gamma t/m}) \right)
    • For t≫τpt \gg \tau_p (persistence time, τp=m/γ\tau_p = m/\gamma): ⟨Δx2(t)⟩≈2kBTγt\langle \Delta x^2(t) \rangle \approx \frac{2k_BT}{\gamma} t
      • Exhibits diffusive behavior
      • Characterized by the D=kBTγD = \frac{k_BT}{\gamma}
  • (VACF): Cv(t)=⟨v(t)v(0)⟩C_v(t) = \langle v(t) v(0) \rangle
    • Measures the correlation between velocities at different times
    • Solution: Cv(t)=kBTme−γt/mC_v(t) = \frac{k_BT}{m} e^{-\gamma t/m}
    • Decays exponentially with characteristic time τp=m/γ\tau_p = m/\gamma
      • Indicates the timescale over which velocity correlations persist

Applications in physical systems

  • Colloidal suspensions:
    • Particles exhibit Brownian motion due to collisions with solvent molecules (water, ethanol)
    • MSD and VACF can be measured using techniques like dynamic light scattering
    • Langevin equation helps interpret experimental data and extract system properties
      • Particle size, solvent viscosity
  • Polymer dynamics:
    • Monomers in a polymer chain undergo Brownian motion
    • : treats the polymer as a chain of beads connected by harmonic springs
      1. Applies Langevin equation to each bead
      2. Includes additional spring forces between beads
    • : includes hydrodynamic interactions between monomers
      1. Modifies the drag coefficient and random force in the Langevin equation
      2. Accounts for the influence of the solvent flow
    • Models predict properties like the diffusion coefficient and relaxation times of polymers
      • Helps understand the viscoelastic behavior of polymer solutions (shear thinning, shear thickening)

Key Terms to Review (21)

Albert Einstein: Albert Einstein was a theoretical physicist best known for developing the theory of relativity, which transformed our understanding of space, time, and gravity. His work laid the foundation for modern physics and has had profound implications across various fields, influencing both quantum mechanics and the study of Brownian motion as well as reshaping ideas about how the universe operates.
Brownian Motion: Brownian motion is the random, erratic movement of microscopic particles suspended in a fluid, resulting from collisions with the fast-moving molecules of the fluid. This phenomenon illustrates the principles of statistical mechanics and plays a vital role in understanding diffusion processes. It serves as a key example of a Markov process, where future states depend only on the present state, and has significant implications in various fields such as physics, finance, and biology.
Brownian Path: A Brownian path refers to the random trajectory traced by a particle undergoing Brownian motion, which is characterized by continuous and erratic movements due to collisions with surrounding fluid molecules. This concept is central to understanding the stochastic nature of particle dynamics and is essential in deriving equations like the Langevin equation, which describes the motion of particles in a fluid under the influence of both deterministic and random forces.
Colloidal suspensions: Colloidal suspensions are mixtures where tiny particles are dispersed throughout a continuous medium, typically liquid, without settling out. These particles, ranging in size from 1 nanometer to 1 micrometer, remain suspended due to Brownian motion, which results from the random thermal agitation of molecules in the surrounding fluid. This unique property of colloidal suspensions leads to interesting phenomena, such as Tyndall effect and stability issues, making them essential in various scientific and industrial applications.
Diffusion Coefficient: The diffusion coefficient is a parameter that quantifies the rate at which particles, such as atoms or molecules, spread out from an area of higher concentration to an area of lower concentration. It plays a critical role in understanding processes like Brownian motion, where particles undergo random motion due to collisions with other particles, and is integral to the Langevin equation, which describes the dynamics of such systems in terms of forces and fluctuations.
Fluctuations: Fluctuations refer to the random and often short-term variations in a physical quantity around its average value. These variations can arise from thermal energy, external forces, or intrinsic properties of the system, and they play a crucial role in understanding dynamic processes at microscopic and macroscopic levels.
Kinetic Theory: Kinetic theory is a scientific theory that explains the behavior of gases in terms of the motion of their individual particles. It connects the macroscopic properties of gases, such as pressure and temperature, to the microscopic behavior of molecules, which are in constant random motion and collide with each other and the walls of their container. This theory provides a foundation for understanding phenomena like Brownian motion and is essential for deriving equations like the Langevin equation that describe particle dynamics under random forces.
Langevin Equation: The Langevin equation is a stochastic differential equation that describes the dynamics of a particle in a fluid, incorporating both deterministic forces and random forces due to thermal fluctuations. This equation is pivotal in modeling Brownian motion, linking macroscopic physical phenomena with microscopic random processes. It serves as a bridge between Markov processes and the statistical mechanics of particles in motion.
Mean-square displacement: Mean-square displacement (MSD) is a statistical measure used to quantify the average squared distance that particles move from their original position over time. It is a key concept in understanding the dynamics of particles in various systems, particularly in contexts such as Brownian motion, where it reflects how particles explore their environment due to random thermal fluctuations and forces.
Perrin: Perrin refers to a concept and mathematical model that describes the behavior of particles undergoing Brownian motion, particularly in relation to the statistical properties of their displacement over time. This model is often connected to the Langevin equation, which provides a framework for understanding the dynamics of particles influenced by random forces, revealing insights into how microscopic movements can lead to macroscopic phenomena such as diffusion.
Polymer dynamics: Polymer dynamics refers to the study of the motion and behavior of polymer chains, particularly in relation to their conformational changes and interactions with their environment. This field examines how factors such as temperature, concentration, and molecular weight affect the movements and arrangements of polymers. Understanding polymer dynamics is crucial for applications in materials science, biology, and nanotechnology, especially when analyzing how polymers respond to forces and external conditions.
Random force: A random force is an unpredictable and fluctuating force that acts on a particle or system, often resulting from the influence of surrounding particles or environmental factors. This concept is crucial in understanding phenomena like Brownian motion, where particles experience random collisions that lead to erratic movements. In the context of stochastic processes, these forces contribute to the system's dynamics by introducing a level of randomness that can be mathematically modeled using equations such as the Langevin equation.
Robert Brown: Robert Brown was a Scottish botanist and naturalist best known for his discovery of Brownian motion, which describes the random movement of particles suspended in a fluid. His observations of pollen grains moving in water laid the foundation for understanding microscopic phenomena and contributed significantly to the field of statistical mechanics and thermodynamics.
Rouse Model: The Rouse model is a theoretical framework used to describe the dynamics of polymer chains in solution, particularly focusing on the diffusion and relaxation processes of these chains. It accounts for the effects of thermal fluctuations and hydrodynamic interactions, providing a way to understand how polymers behave at a molecular level under various conditions. This model is instrumental in explaining phenomena such as Brownian motion, which highlights the random movement of particles suspended in a fluid, and is relevant when discussing the Langevin equation, which describes the motion of these particles subject to random forces.
Statistical mechanics: Statistical mechanics is a branch of theoretical physics that applies probability theory to study the behavior of large ensembles of particles, allowing for the prediction of thermodynamic properties from microscopic properties. This approach bridges the gap between microscopic interactions of individual particles and macroscopic observations, making it essential for understanding phenomena in various fields such as thermodynamics, quantum mechanics, and materials science.
Stochastic process: A stochastic process is a collection of random variables representing a system that evolves over time according to probabilistic rules. It models the uncertainty and randomness inherent in systems influenced by random factors, making it crucial for analyzing phenomena in various fields, including physics, finance, and biology. This concept is especially important for understanding phenomena like Brownian motion, which describes the random movement of particles in fluid, and serves as a foundation for the Langevin equation that describes the dynamics of these systems.
Thermal noise: Thermal noise, also known as Johnson-Nyquist noise, is the random electrical noise generated by the thermal agitation of charge carriers (usually electrons) in a conductor at equilibrium. This type of noise is a fundamental phenomenon observed in all electrical circuits and has significant implications in the study of Brownian motion and the Langevin equation, as it contributes to the stochastic behavior of particles suspended in a fluid.
Velocity Autocorrelation Function: The velocity autocorrelation function measures how the velocity of a particle at one time is correlated with its velocity at another time, providing insight into the dynamics of particle motion in fluids or gases. It is a critical concept in understanding Brownian motion and is often used in the context of the Langevin equation to describe the random motion of particles influenced by collisions with surrounding molecules. This function helps to quantify the persistence of particle velocities and is essential for characterizing diffusion processes.
Viscous drag: Viscous drag is the force that opposes the motion of an object through a fluid due to the fluid's viscosity. This force plays a crucial role in determining how particles behave in fluids, particularly at small scales, where the effects of viscosity are more pronounced compared to inertial forces. Understanding viscous drag is essential for analyzing phenomena like Brownian motion, where tiny particles experience random collisions with fluid molecules, impacting their motion and diffusion.
Wiener Process: A Wiener process is a mathematical representation of continuous-time stochastic processes, specifically a type of Brownian motion. It is characterized by its properties of having independent increments, stationary increments, and being continuous with respect to time, which makes it fundamental in modeling random phenomena in physics and finance. The Wiener process serves as a cornerstone in stochastic calculus and is crucial for formulating the Langevin equation that describes the dynamics of particles subject to random forces.
Zimm Model: The Zimm Model is a theoretical framework used to describe the behavior of polymers in solution, particularly focusing on their dynamics and interactions during Brownian motion. This model effectively connects the macroscopic properties of polymers to their microscopic behavior, explaining how the random movement of polymer chains influences their overall conformation and characteristics in a solvent.
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