2.1 Background
35
where k B is a universal constant currently known as the Boltzmann constant and W
the number of microscopic states. Equation 2.8 is known as Boltzmann’s principle
or Boltzmann’s relation. Note that its left-hand side is a macroscopic quantity while
the right-hand side is calculated using a microscopic quantity. This simple equation
bridges the macroscopic and microscopic worlds.
Although the physical basis of Boltzmann’s principle (Eq. 2.8) remains under
extensive study, there is no doubt about its validity (adequacy) for macroscopic
systems. Then, it offers a novel way to determine the number of microscopic states.
For example, suppose a system consisting of particles, each of which has two internal
states. We distinguish two states by assigning σ = ±1. When the interaction among
them is of two-particle type, the interaction may be expressed as −J i j σ i σ j . The total
energy is given by
−
i
j =i
J i j σ i σ j .
(2.9)
This model assigning two-state to each particle is known as the Ising model [4, 5],
which will be roughly analyzed in Sect. 6.2.1. The model exhibits a phase transition
under some conditions but does not under other conditions. Therefore, its thermodynamic properties show a wide variety. However, according to Boltzmann’s principle,
the entropy of the system is N k B ln 2 at a sufficiently high temperature irrespective of
the setting of {J i j }. More precisely, S ≈ N k B ln 2 at T |J i j | max . It is astonishing.
Since the entropy of transition is expressed in terms of the number of microscopic
states of two phases, W H and W L , as
Δ trs s = k B ln W H − k B ln W L
= k B ln
W H
W L
,
(2.10)
we can determine the change in the number of microscopic states occurring in the
phase transition without referring microscopic details of the substance.
4 If the transition is of purely first-order without any excess heat capacities on both (upper and
lower) sides of the transition temperature, Δ trs s can be regarded as the excess entropy
involved in the transition (Δ ex s). On the other hand, even in the case of continuous
transitions such as one exhibited by the two-dimensional Ising model [6], we need
to estimate Δ ex s for analyzing the transition mechanism.
Since the classical thermodynamics applies to systems with any complexity, Boltzmann’s principle offers a novel strategy of microscopic study of complex systems,
including molecular systems, if the entropy of real systems is experimentally available. Its experimental determination is performed, after Clausius, through
d S =
δq
T
,
(2.11)
4 Although Eq. 2.10 always holds, it depends on problems whether such an interpretation is scientifically fruitful or not. For example, the entropy of fusion is generally harder to rationalize than the
entropy of spin systems like the Ising model.
35
where k B is a universal constant currently known as the Boltzmann constant and W
the number of microscopic states. Equation 2.8 is known as Boltzmann’s principle
or Boltzmann’s relation. Note that its left-hand side is a macroscopic quantity while
the right-hand side is calculated using a microscopic quantity. This simple equation
bridges the macroscopic and microscopic worlds.
Although the physical basis of Boltzmann’s principle (Eq. 2.8) remains under
extensive study, there is no doubt about its validity (adequacy) for macroscopic
systems. Then, it offers a novel way to determine the number of microscopic states.
For example, suppose a system consisting of particles, each of which has two internal
states. We distinguish two states by assigning σ = ±1. When the interaction among
them is of two-particle type, the interaction may be expressed as −J i j σ i σ j . The total
energy is given by
−
i
j =i
J i j σ i σ j .
(2.9)
This model assigning two-state to each particle is known as the Ising model [4, 5],
which will be roughly analyzed in Sect. 6.2.1. The model exhibits a phase transition
under some conditions but does not under other conditions. Therefore, its thermodynamic properties show a wide variety. However, according to Boltzmann’s principle,
the entropy of the system is N k B ln 2 at a sufficiently high temperature irrespective of
the setting of {J i j }. More precisely, S ≈ N k B ln 2 at T |J i j | max . It is astonishing.
Since the entropy of transition is expressed in terms of the number of microscopic
states of two phases, W H and W L , as
Δ trs s = k B ln W H − k B ln W L
= k B ln
W H
W L
,
(2.10)
we can determine the change in the number of microscopic states occurring in the
phase transition without referring microscopic details of the substance.
4 If the transition is of purely first-order without any excess heat capacities on both (upper and
lower) sides of the transition temperature, Δ trs s can be regarded as the excess entropy
involved in the transition (Δ ex s). On the other hand, even in the case of continuous
transitions such as one exhibited by the two-dimensional Ising model [6], we need
to estimate Δ ex s for analyzing the transition mechanism.
Since the classical thermodynamics applies to systems with any complexity, Boltzmann’s principle offers a novel strategy of microscopic study of complex systems,
including molecular systems, if the entropy of real systems is experimentally available. Its experimental determination is performed, after Clausius, through
d S =
δq
T
,
(2.11)
4 Although Eq. 2.10 always holds, it depends on problems whether such an interpretation is scientifically fruitful or not. For example, the entropy of fusion is generally harder to rationalize than the
entropy of spin systems like the Ising model.