statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...

and

thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...

the compressibility equation refers to an equation which relates the isothermal

compressibility In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...

(and indirectly the pressure) to the structure of the liquid. It reads:

kT\left(\frac\right)=1+\rho \int_V \mathrm \mathbf (r)-1

where

\rho

is the number density, g(r) is the

radial distribution function In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If ...

and

kT\left(\frac\right)

is the isothermal

. Using the Fourier representation of the Ornstein-Zernike equation the compressibility equation can be rewritten in the form:

\frac\left(\frac\right) = \frac=\frac=1-\rho\hat(0)=1-\rho \int c(r) \mathrm \mathbf

where h(r) and c(r) are the indirect and direct correlation functions respectively. The compressibility equation is one of the many

integral equations In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...

References

#D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976 Statistical mechanics Thermodynamic equations {{thermodynamics-stub