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10 The Statistical Basis of Bioenergetics
10.2 Statistical Basis of Thermodynamics
With the ideas of Ludwig Boltzmann in 1877, we can now understand the second
law of thermodynamics from the statistics of a large number of interacting particles,
and the observation that exact information about most systems containing many
interacting subsystems is practically lost after a characteristic relaxation time. The
classical statements of the second law of thermodynamics using the imperatives
“must” and “never” become statistical statements with the phrases from probability
language, “very likely” and “very unlikely”. 2
The idea that systems become microscopically intractable introduces a disorder
based on our reduced knowledge about such a system. We replace certainty by
ascribing probability to the possible configurations. 3 If nature followed Newton’s
laws exactly, then with big enough computers we should be able to track the throw
of dice, given an exact measure of their initial position and motion when launched.
Newtonian theory is ‘deterministic’. In practice, when we compute physical values
with numbers assumed to have values over a continuum (rather than discrete or
purely rational), we must truncate those numbers after a certain number of digits. In
following many complex dynamical systems, the error introduced by truncation can
grow exponentially with each iteration. But for macroscopic systems which have
‘settled down’, quantities such as temperature and pressure, which are averages
over microscopic properties, should be measurable, and relationships between
these quantities may be simple, even when the microscopic behavior might be
computationally intractable and impractical.
Statistics is applied to macroscopic systems by assigning probabilities to configurations of possible microstates. Thermal equilibrium occurs when the system
reaches a quasi-static state, spending most of the time in its most probable states. If
we use quantum theory to describe systems, then the properties of that system can
be formulated in terms of the wave function for each possible state of the system.
The wave function, often called is a complex number defined throughout
space within small volumes surrounding possible locations of each particle for the
system. The wave function is used to calculate the probability that each particle
will be found in its given small volume by finding its magnitude squared and then
multiplying by the small volumes. Quantum theory shows how, in principle, wave
functions can be calculated. Quantum theory introduces a new aspect of nature: If a
particle can be found in several distinct quantum states, the wave functions for those
states add to find the overall wave function, and, when calculating probabilities, the
2 For good descriptions of the statistical basis of thermodynamics, see W.J. Moore, Physical
Chemistry, 5th Ed., [Longman, London] (1960) and N. Davidson, Statistical Mechanics,
[McGraw Hill, New York] (1962).
3 In fact, standard probability theory requires the concept of random numbers. But the very tests for
random numbers require an infinite number of trials, so that the application of probabilities also
has a practical difficulty. We can make an increasing larger number of tests, and thereby become
more and more confident in the predictions of a probability theory.
10 The Statistical Basis of Bioenergetics
10.2 Statistical Basis of Thermodynamics
With the ideas of Ludwig Boltzmann in 1877, we can now understand the second
law of thermodynamics from the statistics of a large number of interacting particles,
and the observation that exact information about most systems containing many
interacting subsystems is practically lost after a characteristic relaxation time. The
classical statements of the second law of thermodynamics using the imperatives
“must” and “never” become statistical statements with the phrases from probability
language, “very likely” and “very unlikely”. 2
The idea that systems become microscopically intractable introduces a disorder
based on our reduced knowledge about such a system. We replace certainty by
ascribing probability to the possible configurations. 3 If nature followed Newton’s
laws exactly, then with big enough computers we should be able to track the throw
of dice, given an exact measure of their initial position and motion when launched.
Newtonian theory is ‘deterministic’. In practice, when we compute physical values
with numbers assumed to have values over a continuum (rather than discrete or
purely rational), we must truncate those numbers after a certain number of digits. In
following many complex dynamical systems, the error introduced by truncation can
grow exponentially with each iteration. But for macroscopic systems which have
‘settled down’, quantities such as temperature and pressure, which are averages
over microscopic properties, should be measurable, and relationships between
these quantities may be simple, even when the microscopic behavior might be
computationally intractable and impractical.
Statistics is applied to macroscopic systems by assigning probabilities to configurations of possible microstates. Thermal equilibrium occurs when the system
reaches a quasi-static state, spending most of the time in its most probable states. If
we use quantum theory to describe systems, then the properties of that system can
be formulated in terms of the wave function for each possible state of the system.
The wave function, often called is a complex number defined throughout
space within small volumes surrounding possible locations of each particle for the
system. The wave function is used to calculate the probability that each particle
will be found in its given small volume by finding its magnitude squared and then
multiplying by the small volumes. Quantum theory shows how, in principle, wave
functions can be calculated. Quantum theory introduces a new aspect of nature: If a
particle can be found in several distinct quantum states, the wave functions for those
states add to find the overall wave function, and, when calculating probabilities, the
2 For good descriptions of the statistical basis of thermodynamics, see W.J. Moore, Physical
Chemistry, 5th Ed., [Longman, London] (1960) and N. Davidson, Statistical Mechanics,
[McGraw Hill, New York] (1962).
3 In fact, standard probability theory requires the concept of random numbers. But the very tests for
random numbers require an infinite number of trials, so that the application of probabilities also
has a practical difficulty. We can make an increasing larger number of tests, and thereby become
more and more confident in the predictions of a probability theory.