Temperature

(176) |

It follows from energy conservation that

(177) |

(178) | |||

(179) |

The conservation of energy then reduces to

It is clear that if the systems and are suddenly
brought into thermal contact then they will only exchange heat and evolve towards a
new equilibrium state if the final state is
more probable than the initial one. In other words, if

(181) |

(182) |

(183) |

(184) |

where , , and use has been made of Eq. (180).

It is clear, from the above, that the parameter , defined

*If two systems separately in equilibrium have the same value of then the systems will remain in equilibrium when brought into thermal contact with one another.**If two systems separately in equilibrium have different values of then the systems will not remain in equilibrium when brought into thermal contact with one another. Instead, the system with the**higher*value of will*absorb*heat from the other system until the two values are the same [see Eq. (185)].

Let us define the dimensionless parameter , such that

If two systems are separately in thermal equilibrium with a third system then they must also be in thermal equilibrium with one another.

The thermodynamic temperature of a macroscopic body, as defined in Eq. (187),
depends
only on the rate of change of
the number of accessible microstates with the total energy. Thus, it is possible
to define a thermodynamic
temperature for systems with radically different microscopic
structures (*e.g.*, matter and radiation).
The thermodynamic, or *absolute*, scale of temperature is measured in
degrees *kelvin*.
The parameter is chosen to make this temperature scale
accord as much as possible
with more conventional temperature scales. The choice

(188) |

The familiar
scaling for translational degrees of freedom yields

(189) |

The absolute
temperature is usually positive,
since
is ordinarily a very rapidly increasing function of
energy.
In fact, this is the case for all conventional systems
where the kinetic energy of the particles is taken into account, because there is
no upper bound on the possible energy of the system,
and
consequently increases
roughly like . It is, however, possible to envisage a situation in which we
ignore the translational degrees of freedom of a system, and concentrate only
on its *spin* degrees of freedom. In this case, there is an upper bound to
the possible energy of the system (*i.e.*, all spins lined up anti-parallel to
an applied magnetic field). Consequently, the total number of states available to
the system is finite. In this situation, the density of spin states
first increases with increasing energy, as in conventional
systems, but then reaches a maximum and decreases again. Thus, it is possible to
get absolute spin temperatures which are *negative*, as well as positive.

In Lavoisier's calorific theory, the basic mechanism which forces heat
to flow from hot to cold bodies is the supposed mutual repulsion of the constituent
particles of calorific fluid. In statistical mechanics,
the explanation is far less contrived. Heat flow occurs
because statistical systems tend to evolve towards their most
probable states, subject to the imposed physical constraints.
When two bodies at different temperatures are suddenly
placed in thermal contact, the initial state
corresponds to a spectacularly improbable state of the overall system. For systems
containing of order 1 mole of particles, the only reasonably probable final
equilibrium
states are such that the two bodies differ in temperature by less than 1 part in .
The evolution of the system towards these final states
(*i.e.*, towards thermal equilibrium) is effectively driven by
*probability*.