The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy ; and Shannon's noisy-channel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold, called the channel capacity.

Information rate is the average entropy per symbol.

It is common in information theory to speak of the " rate " or " entropy " of a language.

The entropy rate for the coin is one bit per toss.

This is studied in chemical thermodynamics, which sets limits on quantities like how far a reaction can proceed, or how much energy can be converted into work in a combustion engine and which provides links between properties like the thermal expansion coefficient and change of rate in entropy with pressure for a gas or a liquid.

He also introduced the concept of the chemical potential of a given chemical species, defined to be the rate of the increase in U associated with the increase in the number N of molecules of that species ( at constant entropy and volume ).

* The use of an additional entropy coding tool, and higher frequency accuracy ( due to the larger number of frequency sub-bands used by MP3 ) explains why MP3 doesn't need as high a bit rate as MP2 to get an acceptable audio quality.

They produce entropy at a finite rate.

In a process in which the temperature is locally continuously defined, the local density of rate of entropy production times local temperature gives the local density of dissipated power.

The Lyapunov spectrum can be used to give an estimate of the rate of entropy production

To quantify predictability, the rate of divergence of system trajectories in phase space can be measured ( Kolmogorov-Sinai entropy, Lyapunov exponents ).

where denotes the process limited to duration, and or simply denotes the entropy rate of, which must exist for all discrete-time stationary processes including the ergodic ones.

A theoretical lower bound for the encoding bit rate for lossless data compression is the source information rate, also known as the entropy rate.

* Kolmogorov – Sinai entropy, the rate of information generation by a measure-preserving dynamical system

* Volume entropy, a Riemannian invariant measuring the exponential rate of volume growth of a Riemannian metric

The use of sufficiently well-designed entropy coding techniques can result in the use of a bit rate that is close to the true information content of the indices, such that effectively

In the above formulation, if the bit rate constraint is neglected by setting equal to 0, or equivalently if it is assumed that a fixed-length code ( FLC ) will be used to represent the quantized data instead of a variable-length code ( or some other entropy coding technology such as arithmetic coding that is better than an FLC in the rate – distortion sense ), the optimization problem reduces to minimization of distortion alone.

This is untrue, as consideration of the entropy production in a viscoelastic fluid subject to a sinusoidal time dependent shear rate shows.

Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy ( Rayleigh 1873, Onsager 1931, also ), time rate of entropy production ( Onsager 1931 ), thermodynamic fields, dissipative structure, and non-linear dynamical structure.

In stationary conditions, such forces and associated flux densities are by definition time invariant, as also are the system's locally defined entropy and rate of entropy production.

English text has fairly low entropy.

Uncompressed, English text has about one bit of entropy for each character ( commonly encoded as eight bits ) of message.

Considering that the entropy of written English is less than 1. 1 bits per character, passphrases can be relatively weak.

Another possibility is to use a key text that has more entropy per character than typical English.

Variable-length codes are especially useful when clear text characters have different probabilities ; see also entropy encoding.

A disadvantage of text files is that they usually have a low entropy, meaning that the information occupies more storage than is strictly necessary.

Later in the same text, Maxwell wrote: " Hence the result of the conduction and radiation of heat from one part of a system to another is to diminish the entropy of the system, or the energy, available as work, which can be obtained from the system.

In contrast with what Maxwell wrote then, Tait had changed his mind by 1884 when in his text he accepted Clausius's original definition of entropy.

The increased size may be countered by lower-level link compression, as the resulting text data will have about as much less entropy as it has increased size, so the actual data transferred in this scenario would likely be very close to the size of the original binary data.

There is a well-known relationship between probability and entropy which states that Af, where **zq is the probability that state ( i.e., volume for an ideal gas ) could be reached by chance alone.

This is known as conformational entropy.

This conformational entropy is, in this case, equal to the usual entropy, for there are no other changes or other energies involved.

More formally, it is the theoretical temperature at which entropy reaches its minimum value.

The implication is that the entropy of a perfect crystal simply approaches a constant value.

The question remains whether both can have zero entropy at T = 0 even though each is perfectly ordered.

Since the relation between changes in Gibbs free energy ( G ), the enthalpy ( H ) and the entropy is

Because temperature is defined by the relationship between energy and entropy, such a system's temperature becomes negative, even though energy is being added.

In an adiabatic irreversible process, dQ = 0 is not equal to TdS ( TdS > 0 ) where Q is thermal energy, T is temperature, and S is entropy.

Conversely, an adiabatic process that is irreversible and extracts no work is in an isenthalpic process, such as viscous drag, progressing towards a nonnegative change in entropy.

Since temperature is thermodynamically conjugate to entropy, the isothermal process is conjugate to the adiabatic process for reversible transformations.

The mathematical equation for an ideal gas undergoing a reversible ( i. e., no entropy generation ) adiabatic process is

Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible.

An adiabat is a curve of constant entropy on the P-V diagram.