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Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

Central-Force Motion

Central-Force Motion

The central force system operates by exerting a force on an object directed towards a fixed point, typically the origin, with the force magnitude determined by the object's distance from this fixed point. In the context of an object with mass 'm,' polar coordinates are employed to express the equation of motion. Notably, the azimuthal component of force is nonexistent in this system. A comprehensive rewrite and integration of this equation reveal that the product of the squared radial distance...

Random Variables

Random Variables

A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...

Entropy Change in Reversible Processes

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

Distribution of Molecular Speeds

Distribution of Molecular Speeds

The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This predictable distribution of molecular speeds is known as the Maxwell-Boltzmann distribution. The distribution of molecular speeds in liquids is comparable to that of gases but not identical and can help to understand the phenomenon of the boiling and vapor pressure of a liquid. Consider that a molecule requires a...

Gauss's Law: Problem-Solving

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...

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Related Experiment Video

Updated: Jul 19, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

When and why is the random force in Brownian motion a Gaussian process.

P Mazur¹, D Bedeaux

¹Instituut-Lorentz, University of Leiden, P.O. Box 9506, 2300 RA Leiden, the Netherlands.

Biophysical Chemistry

|October 1, 1991

Summary

Causality and time-reversal invariance restrict Langevin equation descriptions of non-linear Markovian systems. A proven theorem shows forces must be Gaussian and white, with systematic forces linked to equilibrium distribution derivatives.

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Area of Science:

Statistical Mechanics
Non-linear Dynamics
Theoretical Physics

Background:

Langevin equations are crucial for modeling stochastic processes in physics.
Describing fluctuations in non-linear Markovian systems presents theoretical challenges.
The interplay of causality and time-reversal invariance is fundamental in physical theories.

Purpose of the Study:

To investigate the constraints imposed by causality and time-reversal invariance on Langevin equation descriptions.
To determine the properties of Langevin forces under these fundamental assumptions.
To establish conditions for the validity of such descriptions in non-linear systems.

Main Methods:

A theoretical analysis based on the principles of causality and time-reversal invariance.
Mathematical proof of a theorem concerning the nature of Langevin forces.
Examination of a single-variable non-linear Markovian system.

Main Results:

The assumptions of causality and time-reversal invariance severely restrict Langevin equation formulations.
A Langevin force independent of the system's state must be Gaussian and white.
The systematic force must be proportional to the derivative of the logarithm of the equilibrium distribution.

Conclusions:

The applicability of Langevin equations to non-linear Markovian systems is constrained by fundamental physical principles.
A specific form for the systematic force is required for consistency with causality and time-reversal invariance.
The findings provide a deeper understanding of stochastic processes in physics.