Distribution of energy between Macroscopic Systems

According to the principle of equal a priori probability, the probability of the finding a system is directly proportional of the accessible states of the joint system (A*).

\( P(E) = C\,\Omega^*(E) \) (1)

where the normalisation constant \(C\) is independent of \(E\).

Let \(\Omega(E)\) and \(\Omega'(E')\) be the numbers of accessible states of systems \(A\) and \(A'\) respectively. Therefore

\( \Omega^*(E) = \Omega(E)\,\Omega'(E') \) (2)

Substituting into equation (1) yields

\( P(E) = C\,\Omega(E)\,\Omega'(E') \) (3)

Taking the natural logarithm of equation (3),

\( \ln P(E) = \ln C + \ln\Omega(E) + \ln\Omega'(E') \) (4)

The most probable value of \(E\), denoted \({\tilde E}\), is the energy at which \(P(E)\) (and therefore \(\ln P(E)\)) attains its maximum:

\( \left.\dfrac{\partial\ln P(E)}{\partial E}\right|_{E={\tilde E}} = 0 \) (5)

Differentiating equation (4) and using \(E' = E^* - E\) (so that \(\partial E'/\partial E = -1\)) gives

\( \dfrac{\partial\ln\Omega(E)}{\partial E} + \dfrac{\partial\ln\Omega'(E')}{\partial E'} \cdot(-1) = 0 \)

or

\( \dfrac{\partial\ln\Omega(E)}{\partial E} = \dfrac{\partial\ln\Omega'(E')}{\partial E'} \) (6)

Defining the quantities

\( \beta(E) \equiv \dfrac{\partial\ln\Omega(E)}{\partial E}, \qquad \beta'(E') \equiv \dfrac{\partial\ln\Omega'(E')}{\partial E'} \) (7)–(8)

we obtain the fundamental equilibrium condition for thermal interaction:

\( \beta({\tilde E}) = \beta'({\tilde E}') \) (equilibrium)

That is, two systems in thermal contact reach equilibrium when their \(\beta\)-parameters become equal. The energy \({\tilde E}\) that satisfies this equality is the most probable energy of system \(A\).

Definition and cencept of Temperature

The quantity \(\beta\) defined in equation (7) has the dimension of inverse energy.

\( \dfrac{1}{\beta} = k' \) (9)

where k' is a quantity, whose dimension is equivalent to the energy

It is therefore natural to introduce a new intensive parameter \(T\) (called the absolute temperature) through the relation

\( \dfrac{1}{\beta} = kT \) (10)

where \(k\) is a universal positive constant known as the Boltzmann constant. Equation (12) can also be written

\( \beta = \dfrac{1}{kT} \)

With this definition the equilibrium condition \(\beta = \beta'\) becomes the familiar statement

\( T = T' \)

i.e., two systems in thermal equilibrium have equal temperatures.

Zeroth law of thermodynamics

The zeroth law of thermodynamics states that if two systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.

Quantitative idea of temperature scale (thermodynamical parameter)

Thus ẞ is such a fundamental thermodynamical parameter which remains same in the systems when they are in thermal equilibrium. A dimensionless quantity T = (1/kẞ) related with ẞ is called absolute temperature. Hence if any two thermal systems are in thermal equilibrium on placing them in contact with each other and there is no exchange of heat, then their temperature T will remain same. The devices based on the above mentioned law and by which it is possible to find any two systems will exchange heat or not when they are placed in contact with each other, is called thermometer.