3.1 Error Function from Statistical Mechanics
37
Here, the mass of the particle and the frequency are set as unity for simplicity. If
there are two particles, it is natural to have the Hamiltonian expressed as the sum of
each,
H =
1
2
p 1 (t)
2
+
1
2
x 1 (t)
2
+
1
2
p 2 (t)
2
+
1
2
x 2 (t)
2 .
(3.2)
However, in this case, the two particles are not correlated. Each behaves as a free
particle; the energy, or Hamiltonian, is simply the sum of their energies. Two
particles are correlated, for example, when a force acts between the first particle
and the second particle. If the magnitude of this force is proportional to the distance
between the two particles, the Hamiltonian has an additional term,
= c (x 1 (t) − x 2 (t))
2 .
(3.3)
Expanding the parentheses, we find it contains cx 1 (t)x 2 (t) expressed as the product
of the dynamical degrees of freedom of the two particles. This represents the
correlation between the two. To summarize, correlation is generally indicated
by a term composed of multiplication of dynamical degrees of freedom in the
Hamiltonian. The strength of the correlation is represented by the product term
coefficient c. This coefficient is called a coupling constant.
Now, given a physical system, how can its statistical-mechanical behavior be
written in general? When a physical system is in contact with a thermal bath with
temperature T , the physical system receives energy from the thermal bath, and
its energy distribution takes the form of the Boltzmann distribution (canonical
distribution). The probability P of realizing a state with energy E is given by
P =
1
Z
exp
−
E
k B T
.
(3.4)
Here k B is the Boltzmann constant and Z is the partition function,
Z =
all states
exp
−
H
k B T
.
(3.5)
This is to normalize the probability P so that the total probability is 1. For simple
examples, see the end-of-chapter columns in this chapter and in Chap. 5.
After the introduction of the statistical mechanics description of physical systems, let us return to Fig. 3.1, which is an example of supervised data. Since x and d
seem to be correlated, first, as a physical system, we consider statistical mechanics
with x as the external field and d as the dynamical degree of freedom. For the sake
of simplicity, we decided to include only the first-order term in d, and write the
simplest Hamiltonian as
H J,x (d) = −(xJ x + yJ y + J ) d .
(3.6)
37
Here, the mass of the particle and the frequency are set as unity for simplicity. If
there are two particles, it is natural to have the Hamiltonian expressed as the sum of
each,
H =
1
2
p 1 (t)
2
+
1
2
x 1 (t)
2
+
1
2
p 2 (t)
2
+
1
2
x 2 (t)
2 .
(3.2)
However, in this case, the two particles are not correlated. Each behaves as a free
particle; the energy, or Hamiltonian, is simply the sum of their energies. Two
particles are correlated, for example, when a force acts between the first particle
and the second particle. If the magnitude of this force is proportional to the distance
between the two particles, the Hamiltonian has an additional term,
= c (x 1 (t) − x 2 (t))
2 .
(3.3)
Expanding the parentheses, we find it contains cx 1 (t)x 2 (t) expressed as the product
of the dynamical degrees of freedom of the two particles. This represents the
correlation between the two. To summarize, correlation is generally indicated
by a term composed of multiplication of dynamical degrees of freedom in the
Hamiltonian. The strength of the correlation is represented by the product term
coefficient c. This coefficient is called a coupling constant.
Now, given a physical system, how can its statistical-mechanical behavior be
written in general? When a physical system is in contact with a thermal bath with
temperature T , the physical system receives energy from the thermal bath, and
its energy distribution takes the form of the Boltzmann distribution (canonical
distribution). The probability P of realizing a state with energy E is given by
P =
1
Z
exp
−
E
k B T
.
(3.4)
Here k B is the Boltzmann constant and Z is the partition function,
Z =
all states
exp
−
H
k B T
.
(3.5)
This is to normalize the probability P so that the total probability is 1. For simple
examples, see the end-of-chapter columns in this chapter and in Chap. 5.
After the introduction of the statistical mechanics description of physical systems, let us return to Fig. 3.1, which is an example of supervised data. Since x and d
seem to be correlated, first, as a physical system, we consider statistical mechanics
with x as the external field and d as the dynamical degree of freedom. For the sake
of simplicity, we decided to include only the first-order term in d, and write the
simplest Hamiltonian as
H J,x (d) = −(xJ x + yJ y + J ) d .
(3.6)