In statistical mechanics

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, the Rushbrooke inequality relates the critical exponent

Critical exponent

Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...

s of a magnetic system which exhibits a first-order phase transition

Phase transition

A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....

 in the thermodynamic limit

Thermodynamic limit

In thermodynamics, particularly statistical mechanics, the thermodynamic limit is reached as the number of particles in a system, N, approaches infinity...

 for non-zero temperature

Temperature

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

 T.

Since the Helmholtz free energy

Helmholtz free energy

In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

  is extensive, the normalization to free energy per site is given as


The magnetization M per site in the thermodynamic limit

Thermodynamic limit

In thermodynamics, particularly statistical mechanics, the thermodynamic limit is reached as the number of particles in a system, N, approaches infinity...

, depending on the external magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

 H and temperature T is given by


where is the spin at the i-th site, and the magnetic susceptibility

Magnetic susceptibility

In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...

 and specific heat at constant temperature and field are given by, respectively


and

Definitions

The critical exponents and are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows





where


measures the temperature relative to the critical point

Critical point (thermodynamics)

In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...

.

Derivation

For the magnetic analogue of the Maxwell relations

Maxwell relations

Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of...

 for the response functions, the relation


follows, and with thermodynamic stability requiring that , one has


which, under the conditions and the definition of the critical exponents gives


which gives the Rushbrooke inequality


Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.