Markov Chains / Processes| Markov Chains / Processes |
In General > s.a. formulations of quantum mechanics.
* Idea: A process in which a system
evolves through a sequence of steps in some set of possible states, the probability
of it going to a certain state in the next step depending only on the state it is
in (no memory); It is characterized by a transition matrix T such that
Tij ≥ 0 for all
i, j and ∑i
Tij = 1 for all j.
* History: Introduced
by Markov in 1906, who just wanted to show that independence was not
needed for the law of large numbers; An example he considered was the
alternation of consonants and vowels in Pushkin's Eugene Onegin, which
he described as a two-state Markov chain; Soon Poincaré was
studying Markov chains on finite groups to study card shuffling; Today
they are in all applied sciences, from population biology to communication
networks, diffusion models, or social mobility.
@ General references: Revuz 84;
Norris 97 [II];
Brémaud 99;
Baldi et al 02 [and exercises];
Borovkov & Hordijk AAP(04) [normed ergodicity];
Stroock 05;
Lecomte et al JSP(07)cm/06 [thermodynamic formalism];
Rivas et al NJP(10)-a1006 [master equations];
van Casteren 10 [time-dependent strong Markov processes on Polish spaces];
Shiraishi et al PRL(18)-a1802 [speed limit].
@ Markov semigroups: Kolokoltsov JSP(07);
Androulakis & Ziemke JMP(15)-a1406 [quantum Markov semigroups].
@ Non-linear: Frank PLA(08);
Frank PhyA(09) [chaos].
@ Non-equilibrium: Lubashevsky et al PhyA(09) [superstatistical description].
@ Numerical simulations:
Stewart 94;
Berg 04 [Monte Carlo];
Brémaud 08;
Diaconis BAMS(09);
> s.a. montecarlo method.
@ Evolution, examples: Cufaro Petroni & Vigier IJTP(79) [at the speed of light, and the Klein-Gordon equation];
Albeverio & Høegh-Krohn RPMP(84) [fields];
Schächter FP(87);
Ibison CSF(99)qp/01 [1+1 Dirac equation];
Turova JSP(03) [states = directed graphs];
Duchi & Schaeffer JCTA(05) [jumping particles, and Catalan numbers];
Lecomte et al PRL(05) [dynamic partition function, entropy];
Grone et al JPA(08) [reversible, coarse-graining of stochastic matrix];
Hou et al a0805 [and growing networks];
Eliazar JPA(12) [Poissonian steady states];
> s.a. gas [lattice gas].
@ Path-space maximum entropy: Pavon & Ticozzi JMP(10)-a0811;
Lee & Pressé JChemP(12)-a1206 [and n-th order Markov process master equation].
@ Evolution, related topics: Costanza PhyA(11),
PhyA(12) [derivation of deterministic evolution equations];
Cubitt et al PRL(12) [solving the embedding problem];
Jeknić-Dugić et al PRS(16)-a1510 [dynamical emergence of time-coarse-grained Markovianity];
Baez & Courser TAC-a1710 [coarse-graining];
Majid a2002 [quantum geometric interpretation].
> Related topics: see Master
Equation [including generalizations and non-Markovian dynamics]; noether theorem.
> Online resources:
see MathWorld page;
Ryan Ward's page;
Wikipedia page.
Related Processes
> s.a. Martingales; random process [walk].
@ Generalizations: Schreiber JSP(10)-a0905 [polygonal Markov fields];
> s.a. stochastic processes [non-Markovian].
Quantum Markov Processes
> s.a. Adiabaticity; open systems.
@ General references: Dynkin 82;
Ghirardi et al PRA(90);
Marbeau & Gudder AIHP(90);
Gudder & Schindler JMP(91);
Accardi et al mp/04 [for spin chains];
Tay & Petrosky PRA(07)-a0705 [thermal symmetry];
Ibinson et al CMP(08) [robustness];
Leifer & Poulin AP(08)
[quantum graphical models of belief propagation];
Patra & Brooke PRA(08)-a0808 [decoherence-free quantum information];
Gudder JMP(08);
Kraus et al PRA(08)-a0810 [and entanglement production];
Faigle & Schönhuth a1011 [discrete];
Vacchini et al NJP(11)-a1106 [in quantum and classical systems];
Chruściński & Kossakowski JPB(12)-a1201 [Markovianity criteria];
Fannes & Wouters a1204 [fermionic];
Matsumoto a1212 [loss of memory and convergence];
Jeknić-Dugić et al a1905 [no support for the ensemble interpretation];
> s.a. dissipative systems.
@ Non-Markovianity:
Bhattacharya et al a1803 [resource theory].
@ Measures of "Markovianity":
Wolf et al PRL(08);
Haikka et al PRA(11)-a1011;
Alipour et al PRA(12)-a1203 [from quantum discord];
Haseli et al QIP-a1406;
Li et al PRP(18)-a1712 [hierarchy];
> s.a. Loschmidt Echo.
@ Semi-Markov processes:
Breuer & Vacchini PRL(08);
Utagi et al a2012 [non-Markovianity].
@ Other generalizations:
Tarasov TMP(09)-a0909 [fractional];
Brown & Poulin a1206
[Quantum Markov networks, and Gibbs states of Hamiltonians].
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send feedback and suggestions to bombelli at olemiss.edu – modified 26 dec 2020