statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

, the Rushbrooke inequality relates the

critical exponent Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its g ...

s of a

magnetic Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, m ...

system which exhibits a first-order

phase transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...

in the

thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the Limit (mathematics), limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of ...

for non-zero

temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...

''T''. Since the

Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature ( isothermal). The change in the Helmholtz ene ...

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

f = -kT \lim_ \frac\log Z_N

The

magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quanti ...

''M'' per site in the

, depending on the external

magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...

''H'' and temperature ''T'' is given by :

M(T,H) \ \stackrel\   \lim_ \frac \left( \sum_i \sigma_i \right)

where

\sigma_i

is the spin at the i-th site, and the

magnetic susceptibility In electromagnetism, the magnetic susceptibility (; denoted , chi) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the applied magnet ...

and

specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat ...

at constant temperature and field are given by, respectively :

\chi_T(T,H) = \left( \frac \right)_T

and :

c_H = T \left( \frac \right)_H.

Additionally, :

c_M = +T \left( \frac \right)_M.

Definitions

The critical exponents

\alpha, \alpha', \beta, \gamma, \gamma'

and

\delta

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

M(t,0) \simeq (-t)^\mboxt \uparrow 0

M(0,H) \simeq , H, ^ \operatorname(H)\mboxH \rightarrow 0

\chi_T(t,0) \simeq \begin 
	(t)^, & \textrm \ t \downarrow 0 \\
	(-t)^, & \textrm \ t \uparrow 0 \end

c_H(t,0) \simeq \begin
	(t)^ & \textrm \ t \downarrow 0 \\
	(-t)^ & \textrm \ t \uparrow 0 \end

where :

t \ \stackrel\   \frac

measures the temperature relative to the critical point.

Derivation

Using the magnetic analogue of the

Maxwell relations file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant vo ...

for the response functions, the relation :

\chi_T (c_H -c_M) = T \left( \frac \right)_H^2

follows, and with thermodynamic stability requiring that

c_H, c_M\mbox\chi_T \geq 0

, one has :

c_H \geq \frac \left( \frac \right)_H^2

which, under the conditions

H=0, t>0

and the definition of the critical exponents gives :

(-t)^ \geq \mathrm\cdot(-t)^(-t)^

which gives the Rushbrooke inequality :

\alpha' + 2\beta + \gamma' \geq 2.

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

References

{{DEFAULTSORT:Rushbrooke Inequality Critical phenomena Statistical mechanics