The defining expression for entropy in the theory of statistical mechanics established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s, is of the form:
S
=
−
k
B
∑
i
p
i
ln
p
i
,
where
p
i
is the probability of the microstate i taken from an equilibrium ensemble.
The defining expression for entropy in the theory of information established by Claude E. Shannon in 1948 is of the form:
H
=
−
∑
i
p
i
log
b
p
i
,
where
p
i
is the probability of the message
m
i
taken from the message space M and b is the base of the logarithm used. Common values of b are 2, Euler's number e, and 10, and the unit of entropy is shannon (or bit) for b = 2, nat for b = e, and hartley for b = 10.
Mathematically H may also be seen as an average information, taken over the message space, because when a certain message occurs with probability p_{i}, the information quantity −log(p_{i}) will be obtained.
If all the microstates are equiprobable (a microcanonical ensemble), the statistical thermodynamic entropy reduces to the form, as given by Boltzmann,
S
=
k
B
ln
W
,
where W is the number of microstates.
If all the messages are equiprobable, the information entropy reduces to the Hartley entropy
H
=
log
b

M

,
where

M

is the cardinality of the message space M.
The logarithm in the thermodynamic definition is the natural logarithm. It can be shown that the Gibbs entropy formula, with the natural logarithm, reproduces all of the properties of the macroscopic classical thermodynamics of Rudolf Clausius. (See article: Entropy (statistical views)).
The logarithm can also be taken to the natural base in the case of information entropy. This is equivalent to choosing to measure information in nats instead of the usual bits (or more formally, shannons). In practice, information entropy is almost always calculated using base 2 logarithms, but this distinction amounts to nothing other than a change in units. One nat is about 1.44 bits.
For a simple compressible system that can only perform volume work, the first law of thermodynamics becomes
d
E
=
−
p
d
V
+
T
d
S
.
But one can equally well write this equation in terms of what physicists and chemists sometimes call the 'reduced' or dimensionless entropy, σ = S/k, so that
d
E
=
−
p
d
V
+
k
B
T
d
σ
.
Just as S is conjugate to T, so σ is conjugate to k_{B}T (the energy that is characteristic of T on a molecular scale).
Despite the foregoing, there is a difference between the two quantities. The information entropy H can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p_{i} occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p_{i} specifically. The difference is more theoretical than actual, however, because any probability distribution can be approximated arbitrarily closely by some thermodynamic system.
Moreover, a direct connection can be made between the two. If the probabilities in question are the thermodynamic probabilities p_{i}: the (reduced) Gibbs entropy σ can then be seen as simply the amount of Shannon information needed to define the detailed microscopic state of the system, given its macroscopic description. Or, in the words of G. N. Lewis writing about chemical entropy in 1930, "Gain in entropy always means loss of information, and nothing more". To be more concrete, in the discrete case using base two logarithms, the reduced Gibbs entropy is equal to the minimum number of yes–no questions needed to be answered in order to fully specify the microstate, given that we know the macrostate.
Furthermore, the prescription to find the equilibrium distributions of statistical mechanics—such as the Boltzmann distribution—by maximising the Gibbs entropy subject to appropriate constraints (the Gibbs algorithm) can be seen as something not unique to thermodynamics, but as a principle of general relevance in statistical inference, if it is desired to find a maximally uninformative probability distribution, subject to certain constraints on its averages. (These perspectives are explored further in the article Maximum entropy thermodynamics.)
The Shannon entropy in information theory is sometimes expressed in units of bits per symbol. The physical entropy may be on a "per quantity" basis (h) which is called "intensive" entropy instead of the usual total entropy which is called "extensive" entropy. The "shannons" of a message (H) are its total "extensive" information entropy and is h times the number of bits in the message.
A direct and physically real relationship between h and S can be found by assigning a symbol to each microstate that occurs per mole, kilogram, volume, or particle of a homogeneous substance, then calculating the 'h' of these symbols. By theory or by observation, the symbols (microstates) will occur with different probabilities and this will determine h. If there are N moles, kilograms, volumes, or particles of the unit substance, the relationship between h (in bits per unit substance) and physical extensive entropy in nats is:
S
=
k
B
ln
(
2
)
N
h
where ln(2) is the conversion factor from base 2 of Shannon entropy to the natural base e of physical entropy. N h is the amount of information in bits needed to describe the state of a physical system with entropy S. Landauer's principle demonstrates the reality of this by stating the minimum energy E required (and therefore heat Q generated) by an ideally efficient memory change or logic operation by irreversibly erasing or merging N h bits of information will be S times the temperature which is
E
=
Q
=
T
k
B
ln
(
2
)
N
h
where h is in informational bits and E and Q are in physical Joules. This has been experimentally confirmed.
Temperature is a measure of the average kinetic energy per particle in an ideal gas (Kelvins = 2/3*Joules/k_{b}) so the J/K units of k_{b} is fundamentally unitless (Joules/Joules). k_{b} is the conversion factor from energy in 3/2*Kelvins to Joules for an ideal gas. If kinetic energy measurements per particle of an ideal gas were expressed as Joules instead of Kelvins, k_{b} in the above equations would be replaced by 3/2. This shows that S is a true statistical measure of microstates that does not have a fundamental physical unit other than the units of information, in this case "nats", which is just a statement of which logarithm base was chosen by convention.
A physical thought experiment demonstrating how just the possession of information might in principle have thermodynamic consequences was established in 1929 by Leó Szilárd, in a refinement of the famous Maxwell's demon scenario.
Consider Maxwell's setup, but with only a single gas particle in a box. If the supernatural demon knows which half of the box the particle is in (equivalent to a single bit of information), it can close a shutter between the two halves of the box, close a piston unopposed into the empty half of the box, and then extract
k
B
T
ln
2
joules of useful work if the shutter is opened again. The particle can then be left to isothermally expand back to its original equilibrium occupied volume. In just the right circumstances therefore, the possession of a single bit of Shannon information (a single bit of negentropy in Brillouin's term) really does correspond to a reduction in the entropy of the physical system. The global entropy is not decreased, but information to energy conversion is possible.
Using a phasecontrast microscope equipped with a high speed camera connected to a computer, as demon, the principle has been actually demonstrated. In this experiment, information to energy conversion is performed on a Brownian particle by means of feedback control; that is, synchronizing the work given to the particle with the information obtained on its position. Computing energy balances for different feedback protocols, has confirmed that the Jarzynski equality requires a generalization that accounts for the amount of information involved in the feedback.
In fact one can generalise: any information that has a physical representation must somehow be embedded in the statistical mechanical degrees of freedom of a physical system.
Thus, Rolf Landauer argued in 1961, if one were to imagine starting with those degrees of freedom in a thermalised state, there would be a real reduction in thermodynamic entropy if they were then reset to a known state. This can only be achieved under informationpreserving microscopically deterministic dynamics if the uncertainty is somehow dumped somewhere else – i.e. if the entropy of the environment (or the non informationbearing degrees of freedom) is increased by at least an equivalent amount, as required by the Second Law, by gaining an appropriate quantity of heat: specifically kT ln 2 of heat for every 1 bit of randomness erased.
On the other hand, Landauer argued, there is no thermodynamic objection to a logically reversible operation potentially being achieved in a physically reversible way in the system. It is only logically irreversible operations – for example, the erasing of a bit to a known state, or the merging of two computation paths – which must be accompanied by a corresponding entropy increase. When information is physical, all processing of its representations, i.e. generation, encoding, transmission, decoding and interpretation, are natural processes where entropy increases by consumption of free energy.
Applied to the Maxwell's demon/Szilard engine scenario, this suggests that it might be possible to "read" the state of the particle into a computing apparatus with no entropy cost; but only if the apparatus has already been SET into a known state, rather than being in a thermalised state of uncertainty. To SET (or RESET) the apparatus into this state will cost all the entropy that can be saved by knowing the state of Szilard's particle.
Shannon entropy has been related by physicist Léon Brillouin to a concept sometimes called negentropy. In 1953, Brillouin derived a general equation stating that the changing of an information bit value requires at least kT ln(2) energy. This is the same energy as the work Leo Szilard's engine produces in the idealistic case, which in turn equals to the same quantity found by Landauer. In his book, he further explored this problem concluding that any cause of a bit value change (measurement, decision about a yes/no question, erasure, display, etc.) will require the same amount, kT ln(2), of energy. Consequently, acquiring information about a system’s microstates is associated with an entropy production, while erasure yields entropy production only when the bit value is changing. Setting up a bit of information in a subsystem originally in thermal equilibrium results in a local entropy reduction. However, there is no violation of the second law of thermodynamics, according to Brillouin, since a reduction in any local system’s thermodynamic entropy results in an increase in thermodynamic entropy elsewhere. In this way, Brillouin clarified the meaning of negentropy which was considered as controversial because its earlier understanding can yield Carnot efficiency higher than one. Additionally, the relationship between energy and information formulated by Brillouin has been proposed as a connection between the amount of bits that the brain processes and the energy it consumes.
In 2009, Mahulikar & Herwig redefined thermodynamic negentropy as the specific entropy deficit of the dynamically ordered subsystem relative to its surroundings. This definition enabled the formulation of the Negentropy Principle, which is mathematically shown to follow from the 2nd Law of Thermodynamics, during order existence.
Stephen Hawking often speaks of the thermodynamic entropy of black holes in terms of their information content. Do black holes destroy information? It appears that there are deep relations between the entropy of a black hole and information loss See Black hole thermodynamics and Black hole information paradox.
Hirschman showed, cf. Hirschman uncertainty, that Heisenberg's uncertainty principle can be expressed as a particular lower bound on the sum of the classical distribution entropies of the quantum observable probability distributions of a quantum mechanical state, the square of the wavefunction, in coordinate, and also momentum space, when expressed in Planck units. The resulting inequalities provide a tighter bound on the uncertainty relations of Heisenberg.
It is meaningful to assign a "joint entropy", because positions and momenta are quantum conjugate variables and are therefore not jointly observable. Mathematically, they have to be treated as joint distribution. Note that this joint entropy is not equivalent to the Von Neumann entropy, −Tr ρ lnρ = −⟨lnρ⟩. Hirschman's entropy is said to account for the full information content of a mixture of quantum states.
(Dissatisfaction with the Von Neumann entropy from quantum information points of view has been expressed by Stotland, Pomeransky, Bachmat and Cohen, who have introduced a yet different definition of entropy that reflects the inherent uncertainty of quantum mechanical states. This definition allows distinction between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.)
The fluctuation theorem provides a mathematical justification of the second law of thermodynamics under these principles, and precisely defines the limitations of the applicability of that law for systems away from thermodynamic equilibrium.
There are many criticisms of the link between thermodynamic entropy and information entropy, and some are not without merit.
The most common criticism is that information entropy cannot be related to thermodynamic entropy because there is no concept of temperature, energy, or the second law, in the discipline of information entropy. This can best be discussed by considering the fundamental equation of thermodynamics:
d
U
=
∑
F
i
d
x
i
where the F_{i} are "generalized forces" and the dx_{i} are "generalized displacements". This is analogous to the mechanical equation dE = F dx where dE is the change in the kinetic energy of an object having been displaced by distance dx under the influence of force F. For example, for a simple gas, we have:
d
U
=
T
d
S
−
P
d
V
+
μ
d
N
where the temperature (T ), pressure (P ), and chemical potential (µ ) are generalized forces which, when imbalanced, result in a generalized displacement in entropy (S ), volume (V ) and quantity (N ) respectively, and the products of the forces and displacements yield the change in the internal energy (dU ) of the gas.
In the mechanical example, to declare that dx is not a geometric displacement because it ignores the dynamic relationship between displacement, force, and energy is not correct. Displacement, as a concept in geometry, does not require the concepts of energy and force for its definition, and so one might expect that entropy may not require the concepts of energy and temperature for its definition. The situation is not that simple, however. In classical thermodynamics, which is the study of thermodynamics from a purely empirical, or measurement point of view, thermodynamic entropy can only be measured by considering energy and temperature. Clausius' statement dS= δQ/T, or, equvalently, when all other effective displacements are zero, dS=dU/T, is the only way to actually measure thermodynamic entropy. It is only with the introduction of statistical mechanics, the viewpoint that a thermodynamic system consists of a collection of particles and which explains classical thermodynamics in terms of probability distributions, that the entropy can be considered separately from temperature and energy. This is expressed in Boltzmann's famous entropy formula S=k_{B} ln(W). Here k_{B} is Boltzmann's constant, and W is the number of equally probable microstates which yield a particular thermodynamic state, or macrostate.
Boltzmann's equation is presumed to provide a link between thermodynamic entropy S and information entropy H = −Σi pi ln pi = ln(W) where p_{i}=1/W are the equal probabilities of a given microstate. This interpretation has been criticized also. While some say that the equation is merely a unit conversion equation between thermodynamic and information entropy, this is not completely correct. A unit conversion equation will, e.g., change inches to centimeters, and yield two measurements in different units of the same physical quantity (length). Since thermodynamic and information entropy are dimensionally unequal (energy/unit temperature vs. units of information), Boltzmann's equation is more akin to x = c t where x is the distance travelled by a light beam in time t, c being the speed of light. While we cannot say that length x and time t represent the same physical quantity, we can say that, in the case of a light beam, since c is a universal constant, they will provide perfectly accurate measures of each other. (For example, the lightyear is used as a measure of distance). Likewise, in the case of Boltzmann's equation, while we cannot say that thermodynamic entropy S and information entropy H represent the same physical quantity, we can say that, in the case of a thermodynamic system, since k_{B} is a universal constant, they will provide perfectly accurate measures of each other.
The question then remains whether ln(W) is an informationtheoretic quantity. If it is measured in bits, one can say that, given the macrostate, it represents the number of yes/no questions one must ask to determine the microstate, clearly an informationtheoretic concept. Objectors point out that such a process is purely conceptual, and has nothing to do with the measurement of entropy. Then again, the whole of statistical mechanics is purely conceptual, serving only to provide an explanation of the "pure" science of thermodynamics.
Ultimately, the criticism of the link between thermodynamic entropy and information entropy is a matter of terminology, rather than substance. Neither side in the controversy will disagree on the solution to a particular thermodynamic or informationtheoretic problem.
In 1995, Tim Palmer signalled two unwritten assumptions about Shannon's definition of information that may make it inapplicable as such to quantum mechanics:
The supposition that there is such a thing as an observable state (for instance the upper face of a dice or a coin) before the observation begins
The fact that knowing this state does not depend on the order in which observations are made (commutativity)
Anton Zeilinger's and Caslav Brukner's article synthesized and developed these remarks. The socalled Zeilinger's principle suggests that the quantization observed in QM could be bound to information quantization (one cannot observe less than one bit, and what is not observed is by definition "random"). Nevertheless, these claims remain quite controversial. Detailed discussions of the applicability of the Shannon information in quantum mechanics and an argument that Zeilinger's principle cannot explain quantization have been published, that show that Brukner and Zeilinger change, in the middle of the calculation in their article, the numerical values of the probabilities needed to compute the Shannon entropy, so that the calculation makes little sense.
In 2013, a description was published of a two atom version of a Szilárd engine using Quantum discord to generate work from purely quantum information. Refinements in the lower temperature limit were suggested.
Algorithmic cooling is an algorithmic method for transferring heat (or entropy) from some qubits to others or outside the system and into the environment, thus resulting in a cooling effect. This cooling effect may have usages in initializing cold (highly pure) qubits for quantum computation and in increasing polarization of certain spins in nuclear magnetic resonance.