Entropy

:For other senses of the term entropy, see entropy (disambiguation).

Boltzmann's principle

In Boltzmann's definition, entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium, consistent with its macroscopic thermodynamic properties (or macrostate). To understand what microstates and macrostates are, consider the example of a gas in a container. At a microscopic level, the gas consists of a vast number of freely moving atoms, which occasionally collide with one another and with the walls of the container. The microstate of the system is a description of the positions and momenta of all the atoms. In principle, all the physical properties of the system are determined by its microstate. However, because the number of atoms is so large, the motion of individual atoms is mostly irrelevant to the behavior of the system as a whole. Provided the system is in thermodynamic equilibrium, the system can be adequately described by a handful of macroscopic quantities, called "thermodynamic variables": the total energy E, volume V, pressure P, temperature T, and so forth. The macrostate of the system is a description of its thermodynamic variables.

Related Topics:
Thermodynamic equilibrium - Gas - Vast number - Atom - Position - Momenta - Energy - Volume - Pressure - Temperature

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There are three important points to note. Firstly, to specify any one microstate, we need to write down an impractically long list of numbers, whereas specifying a macrostate requires only a few numbers (E, V, etc.) Secondly, macrostates are only defined when the system is in equilibrium; non-equilibrium situations can generally not be described by a small number of variables. For example, if a gas is sloshing around in its container, even a macroscopic description would have to include, e.g., the velocity of the fluid at each different point. Thirdly, more than one microstate can correspond to a single macrostate. In fact, for any given macrostate, there will be a huge number of microstates that are consistent with the given values of E, V, etc.

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We are now ready to provide a definition of entropy. Let Ω be the number of microstates consistent with the given macrostate. The entropy S is defined as

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:S = k ln Omega ,!

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The quantity k is a physical constant known as Boltzmann's constant, which, like the entropy, has units of heat capacity. The logarithm is dimensionless.

Related Topics:
Physical constant - Boltzmann's constant - Heat capacity - Logarithm - Dimensionless

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This postulate, which is known as Boltzmann's principle, may be regarded as the foundation of statistical mechanics, which describes thermodynamic systems using the statistical behaviour of its constituents. It turns out that S is itself a thermodynamic property, just like E or V. Therefore, it acts as a link between the microscopic world and the macroscopic. One important property of S follows readily from the definition: since Ω is a natural number (1,2,3,...), S is either zero or positive (this is a property of the logarithm.)

Related Topics:
Statistical mechanics - Natural number - Logarithm

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Disorder and the second law of thermodynamics

We can view Ω as a measure of the disorder in a system. This is reasonable because what we think of as "ordered" systems tend to have very few configurational possibilities, and "disordered" systems have very many. As an illustration of this idea, consider a set of 100 coins, each of which is either heads up or tails up. The macrostates are specified by the total number of heads and tails, whereas the microstates are specified by the facings of each individual coin. For the macrostates of 100 heads or 100 tails, there is exactly one possible configuration, corresponding to the most "ordered" state in which all the coins are facing the same way. The most "disordered" macrostate consists of 50 heads and 50 tails in any order, for which there are 100891344545564193334812497256 (100 choose 50) possible microstates.

Related Topics:
Coin - Heads up or tails up - 100 choose 50

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Even when a system is entirely isolated from external influences, its microstate is constantly changing. For instance, the particles in a gas are constantly moving, and thus occupy a different position at each moment of time; their momenta are also constantly changing as they collide with each other or with the container walls. Suppose we prepare the system in an artificially highly-ordered equilibrium state. For instance, imagine dividing a container with a partition and placing a gas on one side of the partition, with a vacuum on the other side. If we remove the partition and watch the subsequent behavior of the gas, we will find that its microstate evolves according to some chaotic and unpredictable pattern, and that on average these microstates will correspond to a more disordered macrostate than before. It is possible, but extremely unlikely, for the gas molecules to bounce off one another in such a way that they remain in one half of the container. It is overwhelmingly probable for the gas to spread out to fill the container evenly, which is the new equilibrium macrostate of the system.

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This is an illustration of a principle that we will prove rigorously in a subsequent section, known as the Second Law of Thermodynamics. This states that

Related Topics:
Subsequent section - Second Law of Thermodynamics

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:The total entropy of an isolated system can never decrease.

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Since its discovery, the idea that disorder tends to increase has been the focus of a great deal of thought, some of it confused. A chief point of confusion is the fact that the Second Law applies only to isolated systems. For example, the Earth is not an isolated system because it is constantly receiving energy in the form of sunlight. Nevertheless, it has been pointed out that the universe may be considered an isolated system, so that its total disorder should be constantly increasing. We will discuss the implications of this idea in the section on Entropy and cosmology.

Related Topics:
Earth - Sunlight - Universe - Entropy and cosmology

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It is important to distinguish the meaning of "disorder" in the context of entropy and the colloquial definition, which is a vague term associated with "chaos". The "disorder" to which we refer in this article is a specific, well-defined quantity.

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Counting of microstates

In classical statistical mechanics, the number of microstates is actually uncountably infinite, since the properties of classical systems are continuous. For example, a microstate of a classical ideal gas is specified by the positions and momenta of all the atoms, which range continuously over the real numbers. If we want to define Ω, we have to come up with a method of grouping the microstates together to obtain a countable set. This procedure is known as coarse graining. In the case of the ideal gas, we count two states of an atom as the "same" state if their positions and momenta are within δx and δp of each other. Since the values of δx and δp can be chosen arbitrarily, the entropy is not uniquely defined. It is defined only up to an additive constant. (As we will see, the thermodynamic definition of entropy is also defined only up to a constant.)

Related Topics:
Classical - Uncountably infinite - Real number - Coarse graining - Thermodynamic definition of entropy

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This ambiguity can be resolved with quantum mechanics. The quantum state of a system can be expressed as a superposition of "basis" states, which can be chosen to be energy eigenstates (i.e. eigenstates of the quantum Hamiltonian.) Usually, the quantum states are discrete, even though there may be an infinite number of them. In quantum statistical mechanics, we can take Ω to be the number of energy eigenstates consistent with the thermodynamic properties of the system.

Related Topics:
Quantum mechanics - Quantum state - Eigenstate - Hamiltonian

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An important result, known as Nernst's theorem or the third law of thermodynamics, states that the entropy of a system at zero absolute temperature is a well-defined constant. This is due to the fact that a system at zero temperature exists in its lowest-energy state, or ground state, so that its entropy is determined by the degeneracy of the ground state. Many systems, such as crystal lattices, have a unique ground state, and (since ln(1) = 0) this means that they have zero entropy at absolute zero. Other systems have more than one state with the same, lowest energy, and have a non-vanishing "zero-point entropy". For instance, ordinary ice has a zero-point entropy of 3.41 J/(mol·K), due to the fact that its underlying crystal structure possesses multiple configurations with the same energy (a phenomenon known as geometrical frustration).

Related Topics:
Nernst's theorem - Third law of thermodynamics - Zero absolute temperature - Ground state - Degeneracy - Crystal lattices - Ice - Crystal structure - Geometrical frustration

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