Entropy |

**In General** > s.a. chaos;
MaxEnt (Maximum Entropy Principle); history
of physics; statistical mechanics; thermodynamics.

* __Thermodynamic entropy__:
An extensive thermodynamical function of state which is non-decreasing for
adiabatic transformations (constant in reversible ones); In most systems,
it grows monotonically with *E*; The concept was introduced by
Clausius in the 1860s and the Clausius entropy is defined, up to an
additive constant, by the relation d*S* = d*Q*/*T*.

* __Statistical
interpretation__: A measure of how spread out a classical distribution
function or quantum density *ρ* matrix is; It can be defined in
various ways; For a quantum density matrix *ρ*, the von Neumann
entropy is defined by *S* = –*k*_{B}*
*tr *ρ* ln *ρ*; > s.a. types
of entropies.

* __And classical /
quantum mechanics__: In classical mechanics one assumes entropies are
finite, but if one takes the \(\hbar\) → 0 limit of any quantum entropy
expression, this diverges; So classical thermodynamics is not the
classical limit of quantum statistical mechanics, it knows about quantum
mechanics.

@ __Articles, II__: Han et al cm/97,
AJP(99)jan
[tracing
over degrees of freedom]; Styer AJP(00)dec
[examples
and meaning]; Lieb & Yngvason PT(00)apr
[without
thermodynamics]; Styer AJP(08)oct
[and
evolution]; Salagaram & Chetty AJP(11)nov
[computation].

@ __Books__: Martin & England 81;
Denbigh & Denbigh 85; Leff
& Rex ed-90; Dugdale 96
[II]; Scully & Scully 07
[Maxwell's demon, information,
and the quantum]; Grandy 08 [and
probability]; Ben-Naim 16
[I].

@ __General references__: Wehrl RMP(78);
in
Sorkin PRL(86);
Mackey
RMP(89)
[increasing entropy and ergodicity];
Lavenda et al NCB(95); Leff AJP(96)oct
[and
energy sharing];
Levy math/06
[mathematical
interpretation]; Kozlov & Treshchev TMP(07)
[coarse-grained];
Bernstein a1109
[pedagogical]; Baumgartner FP(14)-a1206
[axiomatic characterization and derived facts]; Zanchini & Beretta Ent(14)-a1403
[general definition]; Phillips TPT(16) [entropy as the spread of energy].

@ __Conceptual__: Leff FP(07)
[language
and interpretation]; Topsoe a0807
[entropy
and truth vs belief]; Timberlake TPT(10)
[statistical interpretation]; Swendsen AJP(11)apr
[definitions and meaning]; Baez et al Ent(11)-a1106
[characterization in terms of information loss]; Davey PhSc(11)
[and probability]; Swendsen FP(12)
[argument for probabilistic definition]; Dieks a1209-in
[thermodynamical vs statistical-mechanical definitions]; Benguigui EJP(13)-a1209
[different definitions]; Caticha M&M-a1412
[entropic inference as a general framework for reasoning under conditions
of uncertainty]; in Oltean et al PRD(16)-a1607 [nature of *S*]; Toffoli Ent-a1705 [and "honest entropy"].

@ __And observation__: Steane NJP(16)-a1510
[determining absolute entropy without quantum theory or the third law].

**Related Concepts and Generalizations** > s.a. Coarse
Graining; dissipation; Entropic
Dynamics; Homogeneity.

@ __And area law__: Mäkelä gq/06-GRF;
Masanes
PRA(09)-a0907
[for
low-energy states].

@ __Other properties__: Kay & Kay JPA(01)mp/98
[strong
subadditivity and volume]; Touchette AJP(08)jan
[(non)-concavity].

@ __Subsystems__: Hotta & Joichi PLA(99)cm
[additivity]; Baumgartner JPA(02)mp/01;
Dupuis et al a1607
[additivity, "accumulation"].

@ __Entropy current__: Bhattacharyya JHEP(14)-a1403,
JHEP(14)
[and the equilibrium partition function];
Banerjee et al JHEP(14)
[non-relativistic fluid].

@ __Beyond equilibrium situations__: Beretta & Zanchini in(11)-a1010;
Beretta & Zanchini a1411
[and few-particle systems];
> s.a. non-equilibrium
thermodynamics.

@ __And time__: Hotke-Page & Page in(92)
[clock
time]; Ben-Naim 16
[no relationship].

@ __Entropy production and arrow of time__: Parrondo et al NJP(09);
Martyushev & Shaiapin Ent(16)-a1605.

@ __Related topics__: Hall PRA(99)qp/98
[geometric,
ensemble volume]; Leff AJP(99)dec
[dimensionless];
Dover PhyA(04)
[and
scaling laws]; Hanel et al PNAS(12)-a1211
[generalized, for systems violating the Shannon-Khinchin axioms]; Lieb
& Yngvason PRS(14)-a1403
[entropy meters and the entropy of non-extensive systems]; Plastino et al
PhyA(14)
[partition function and entropy from thermodynamics]; Chafaï AFST(15)-a1405
[aspects of entropy in different disciplines].

__Related topics__:
see entropy bounds;
types of entropies and
entanglement entropy.

**Systems**

> __Gravity-related__:
see black-hole entropy; gravitational
thermodynamics; horizons.

> __Other systems__:
see casimir effect [negative entropies];
cell complex; cellular
automaton; networks; thermodynamic
systems; tilings.

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send feedback and suggestions to bombelli at olemiss.edu – modified 7
may 2017