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PDF | In this chapter we will review some basic concepts of thermodynamics and statistical mechanics. Furthermore, we will discuss the energy.
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- Principles Statistical Mechanics by Richard Tolman
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The issue of the role of external interference in the apparently spontaneous behavior of what is idealized as an isolated system has been much discussed. Here the existence of special systems such as spin echo systems encountered in nuclear magnetic resonance plays a role in the arguments. This seems to show entropic increase without the kind of interference from the outside that genuinely destroys the initial order implicit in the system. In such a sub-region we would find a world far from equilibrium. Introducing the familiar time-symmetric probabilistic assumptions, it becomes likely that in such a region one finds states of lower entropy in one time direction and states of higher entropy in the other.
Then finish the solution by introducing the other Boltzmann suggestion that what we mean by the future direction of time is fixed as that direction of time in which entropy is increasing. Current cosmology sees quite a different universe than that posited by Boltzmann. As far as we can tell the universe as a whole is in a highly non-equilibrium state with parallel entropic increase into the future everywhere.
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But the structure of the cosmos as we know it allows for an alternative solution to the problem of the origin of time asymmetry in thermodynamics. The universe seems to be spatially expanding, with an origin some tens of billions of years ago in an initial singularity, the Big Bang. Expansion, however, by itself does not provide the time asymmetry needed for thermodynamics, for an expanding universe with static or decreasing entropy is allowed by physics. Indeed, in some cosmological models in which the universe contracts after expanding, it is usually, though not always, assumed that even in contraction entropy continues to increase.
The source of entropic asymmetry is sought, rather, in the physical state of the world at the Big Bang. A world in which matter is distributed with uniformity is one of low entropy. A high entropy state is one in which we find a clustering of matter into dense regions with lots of empty space separating these regions. This deviation from the usual expectation — spatial uniformity as the state of highest entropy — is due to the fact that gravity, unlike the forces governing the interaction of molecules in a gas for example, is a purely attractive force.
One then has the universe as we know it, with its thermally highly non-equilibrium condition.
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If one conditionalizes on that initial low entropy state one then gets, using the time symmetric probabilities of statistical mechanics, a prediction of a universe whose entropy increased in time. One can argue, in a manner tracing back to H. Most of the arguments in the literature that this will be so are flawed, but the inference is reasonable nonetheless. It has also been suggested that if one invokes some underlying statistical dynamic law such as the GRW law noted above , one need not posit a branch system hypothesis in addition to initial low entropy to derive the thermodynamic results.
Indeed, can we apply probabilistic reasoning appropriate for systems in the universe as we know it to an initial state for the universe as a whole? The issues here are reminiscent of the old debates over the teleological argument for the existence of God. The older theory had no probabilistic qualifications to its laws. Conceptually the relationship of older to newer theory is quite complex. Concepts of the older theory volume, pressure, temperature, entropy must be related to the concepts of the newer theory molecular constitution, dynamical concepts governing the motion of the molecular constituents, probabilistic notions characterizing either the states of an individual system or distributions of states over an imagined ensemble of systems subject to some common constraints.
There is, for example, Boltzmann entropy which is the property of a single system defined in terms of the spatial and momentum distribution of its molecules. On the other hand there are the Gibbs'entropies, definable out of the probability distribution over some Gibbsian ensemble of systems. Adding even more complications there is, for example, Gibbs' fine grained entropy which is defined by the ensemble probability alone and is very useful in characterizing equilibrium states and Gibbs' coarse grained entropy whose definition requires some partitioning of the phase space into finite cells as well as the original probability distribution and which is a useful concept in characterizing approach to equilibrium from the ensemble perspective.
In addition to these notions which are measure theoretic in nature, there are topological notions which can play the role of a kind of entropy as well. Nothing in this complexity stands in the way of claiming that statistical mechanics describes the world in a way that explains why thermodynamics works and works as well as it does.
But the complexity of the inter-relationship between the theories should make the philosopher cautious in using this relationship as a well understood and simple paradigm of inter-theoretic reduction. It is of some philosophical interest that the relationship of thermodynamics to statistical mechanics shows some similarity to aspects uncovered in functionalist theories of the mind-body relationship. Consider, for example, the fact that systems of very different physical constitutions say a gas made up of molecules interacting by means of forces on the one hand and on the other hand radiation whose components are energetically coupled wave lengths of light can share thermodynamic features.
They can, for example, be at the same temperature. Physically this means that the two systems, if initially in equilibrium and then energetically coupled, will retain their original equilibrium conditions. The parallel with the claim that a functionally defined mental state a belief, say can be instantiated in a wide variety of physical devices is clear.
We have noted that it was Boltzmann who first suggested that our very concept of the future direction of time was fixed by the direction in time in which entropy was increasing in our part of the universe. We must first ask what the theory is really claiming.
In a sensible version of the theory there is no claim being made to the effect that we find out the time order of events by checking the entropy of systems and taking the later event as the one in which some system has its higher entropy. There are asymmetries of knowledge: We have memories and records of the past, but not of the future. There are asymmetries of determination: We think of causation as going from past through present to future, and not of going the other way round. There are asymmetries of concern: We may regret the past, but we anxiously anticipate the future. The entropic theory in its most plausible formulation is a claim to the effect that we can explain the origin of all of these intuitive asymmetries by referring to fact about the entropic asymmetry of the world.
This can be best understood by looking at the very analogy used by Boltzmann: the gravitational account of up and down. What do we mean by the downward direction at a spatial location?
All of the phenomena by which we intuitively identify the downward direction as the direction in which rocks fall, for example receive an explanation in terms of the spatial direction of the local gravitational force. Even our immediate awareness of which direction is down is explainable in terms of the effect of gravity on the fluid in our semi-circular canals. Nor are we dismayed to be told that in outer space, far from a large gravitating object such as the Earth, there is no such thing as the up-down distinction and no direction of space which is the downward direction.
Similarly the entropic theorist claims that it is the entropic features that explain the intuitive asymmetries noted above, that in regions of the universe in which the entropic asymmetry was counter-directed in time the past-future directions of time would be opposite, and that in a region of the universe without an entropic asymmetry neither direction of time would count as past or as future. The great problem remains in trying to show that the entropic asymmetry is explanatorily adequate to account for all the other asymmetries in the way that the gravitational asymmetry can account for the distinction of up and down.
Despite many interesting contributions to the literature on this, the problem remains unresolved. Most foundational inquiry into statistical mechanics presumes a classical dynamical basis for describing the dynamics of the constituent components of macroscopic systems. But this cannot be correct, of course, since that underlying dynamics must be quantum mechanical. Gibbs was cautious in claiming a simple explanatory role for his ensemble version of statistical mechanics, for example, since it led to notoriously false predictions for such macroscopic features of systems as their specific heat.
It was later realized that the fault here lay not in Gibbs' statistical mechanics but in assuming classical dynamics at the constituent level. Once systems were re-described on the correct quantum mechanical basis the predictive errors disappeared. Naturally changing to a quantum mechanical basis leads to wholesale changes within statistical mechanics. A new notion of phase space with probabilities over it is needed, for example.
Does this mean, though, that the foundational explorations presupposing classical mechanics are now irrelevant? We have already noted that some proposals have been made that seek for the grounding of the very probabilistic nature of statistical mechanics in the fundamentally probabilistic nature of quantum mechanics at the dynamical level, or, rather, in some interpretation of how probability functions in the roots of quantum mechanics.
Principles Statistical Mechanics by Richard Tolman
Even without going that far, though, the shift to a quantum dynamical basis just require some rethinking of subtle issues in the foundational debates. From the earliest days Poincare's Recurrence Theorem was a problem for statistical mechanics. With a classical dynamical basis a reply could be made that while the Theorem held for individual systems of the concern of the theory, it would not necessarily hold of an ensemble of such systems.
In both dynamical frameworks, however, a move to the thermodynamic limit of an infinite number of constituents for a system can eliminate the applicability of the theorem as an objection to the monotonicity of thermodynamic change being unobtainable in statistical mechanics. One of the most striking macroscopic feature of systems is the existence of multiple phases gas, liquid and solid, for example, or diamagnetic and ferromagnetic for another and the transitions between those phases as thermodynamic features such as temperature and pressure or imposed magnetization are varied.
Early work on phase transitions focused on the way in which quantities changed in a non-analytic manner from phase to phase, even though statistical mechanics seemed to show that such non-analytic behavior was impossible, at least for systems with a finite number of constituents. More recently methods have been developed for dealing with some phase transitions that not only supplement the standard explanatory schemes of traditional statistical mechanics, but that also provide insights into the variety of forms scientific explanations can take.
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In some cases the nature of the transition is seen to depend on a few abstract parameters and not on the physical details of the system. What matters are such things as the dimension of the system, the degrees of freedom of the dynamics of the constituents and general limits on the interactions of the constituents with one another such as short and very long range behavior of the relevant forces of interaction.
The trick is to look first at the interactions of nearest constituents. Then one moves to a block of constituents as it relates to nearest similar blocks. One continues this process to the limit of an infinite system and looks for a limit point to the continually rescaled interaction. The explanatory strategy here is quite unlike the usual one encountered in statistical mechanics and shows incisively how the specifics of the physical systems crying out for explanation in science can require the introduction of new methodological ploys if full understanding is to be obtained.
In this the introduction of these renormalization group methods resembles the way in which the need to an atomistic account of macroscopic thermodynamic behavior itself required the new methods of statistical mechanics to be added to the older repertoire of typical dynamical explanations. A comprehensive treatment of the issues from a philosophical perspective is Sklar Of important historical interest is Reichenbach An accessible and up-to-date discussion of the fundamental issues is Albert The possible invocation of a time asymmetric law through the GRW approach is discussed in this book.
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A spirited defense of the initial low entropy approach to time asymmetry is Price Frigg reviews additional philosopher's work on the foundational issues. English translations of many original fundamental papers are in Brush Brush provides an historical treatment of the development of the theory. Two foundational works that are essential are Gibbs and Ehrenfest and Ehrenfest Two works explaining clearly and detail many of the technical aspects of foundational statistical mechanics are Emch and Liu and Toda, Kubo and Saito These two works provide a thorough grounding on quantum statistical mechanics and how it differs from statistical mechanics grounded on classical theory.
An excellent introduction to phase change and renormalization group theory is Batterman Historical Sketch 2. Philosophers on Probability and Statistical Explanation 3. Equilibrium Theory 4. Bragg-Williams approximation. Cooperative phenomena.
Average potential in charged particle systems. Distribution functions in a particle system. Fluctuations and Kinetic Theories. Collision frequency. Boltzmann transport equation. This book provides a series of concise lectures on the fundamental theories of statistical mechanics, carefully chosen examples and a number of problems with complete solutions.
Modern physics has opened the way for a thorough examination of infra-structure of nature and understanding of the properties of matter from an atomistic point of view. Statistical mechanics is an essential bridge between the laws of nature on a microscopic scale and the macroscopic behaviour of matter. A good training in statistical mechanics thus provides a basis for modern physics and is indispensable to any student in physics, chemistry, biophysics and engineering sciences who wishes to work in these rapidly developing scientific and technological fields.
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