Entropy (statistical thermodynamics): Difference between revisions

The concept '''entropy''' was first developed by German physicist [[Rudolf Clausius]] in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In [[statistical mechanics]], [[entropy]] is formulated as a statistical property using [[probability theory]]. The '''statistical entropy''' perspective was introduced in 1870 by Austrian physicist [[Ludwig Boltzmann]], who established a new field of [[physics]] that provided the descriptive linkage between the macroscopic observation of nature and the microscopic view based on the rigorous treatment of large ensembles of microscopic states that constitute [[thermodynamic system]]s.

 

{{main|Boltzmann's entropy formula}}

Ludwig Boltzmann defined entropy as a measure of the number of possible microscopic states (''microstates'') of a system in [[thermodynamic equilibrium]], consistent with its macroscopic thermodynamic properties, which constitute the ''macrostate'' of the system. A useful illustration is the example of a sample of gas contained in a container. The easily measurable parameters volume, pressure, and temperature of the gas describe its macroscopic condition (''state''). At a microscopic level, the gas consists of a vast number of freely moving [[atom]]s or [[molecule]]s, which randomly collide with one another and with the walls of the container. The collisions with the walls produce the macroscopic pressure of the gas, which illustrates the connection between microscopic and macroscopic phenomena.

Equilibrium may be illustrated with a simple example of a drop of food coloring falling into a glass of water. The dye diffuses in a complicated manner, which is difficult to precisely predict. However, after sufficient time has passed, the system reaches a uniform color, a state much easier to describe and explain.

 

 

Boltzmann's entropy describes the system when all the accessible microstates are equally likely. It is the configuration corresponding to the maximum of entropy at equilibrium. The randomness or disorder is maximal, and so is the lack of distinction (or information) of each microstate.

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<math display="block"> [S] = [k_\text{B}] = \mathrm{\frac {J} {K}}</math>

 

The various ensembles used in [[statistical thermodynamics]] are linked to the entropy by the following relations:{{clarify|reason=What are the quantities that are being maintained constant between these different ensembles? Is this relationship only valid in the thermodynamic limit?|date=September 2013}}

<math display="block">S = k_\text{B} \ln \Omega_{\rm mic} = k_\text{B} (\ln Z_{\rm can} + \beta \bar E) = k_\text{B} (\ln \mathcal{Z}_{\rm gr} + \beta (\bar E - \mu \bar N)) </math>

 

== Order through chaos and the second law of thermodynamics ==

 

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.

 

This is an example illustrating the [[second law of thermodynamics]]:

 

Since its discovery, this idea 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]]. In contrast, the [[universe]] may be considered an isolated system, so that its total entropy is constantly increasing. (Needs clarification. See: [[Second law of thermodynamics#cite note-Grandy 151-21]])

 

To avoid coarse graining one can take the entropy as defined by the [[H-theorem#Tolman's_H-theorem|H-theorem]].<ref>{{cite book |isbn=0-486-68455-5|title=Lectures on Gas Theory|last1=Boltzmann|first1=Ludwig|date=January 1995|publisher=Courier Corporation }}</ref>

 

However, 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 [[eigenstate]]s (i.e. eigenstates of the quantum [[Hamiltonian (quantum mechanics)|Hamiltonian]]). Usually, the quantum states are discrete, even though there may be an infinite number of them. For a system with some specified energy ''E'', one takes Ω to be the number of energy eigenstates within a macroscopically small energy range between ''E'' and {{nowrap|''E'' + ''δE''}}. In the [[thermodynamical limit]], the specific entropy becomes independent on the choice of ''δE''.

An important result, known as [[Nernst's theorem]] or the [[third law of thermodynamics]], states that the entropy of a system at [[absolute zero|zero absolute temperature]] is a well-defined constant. This is because a system at zero temperature exists in its lowest-energy state, or [[ground state]], so that its entropy is determined by the [[Hamiltonian (quantum mechanics)|degeneracy]] of the ground state. Many systems, such as [[crystal|crystal lattices]], have a unique ground state, and (since {{nowrap|1=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 {{val|3.41|u=J/(mol⋅K)}}, because its underlying [[crystal structure]] possesses multiple configurations with the same energy (a phenomenon known as [[geometrical frustration]]).

 

 

== See also ==

 

== References ==

 

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