Meta tags:
Headings (most frequently used words):
and, solution, dimensions, phase, in, one, exact, inequality, ising, model, no, transition, dimension, two, neural, onsager, contents, definition, basic, properties, history, historical, significance, applications, numerical, simulation, solutions, see, also, footnotes, references, external, links, discussion, simplifications, connection, to, graph, maximum, cut, questions, correlation, inequalities, transitions, finite, volume, peierls, droplets, kramers, wannier, duality, yang, lee, zeros, magnetism, lattice, gas, neuroscience, spin, glasses, artificial, network, sea, ice, cayley, tree, topologies, large, networks, metropolis, algorithm, as, markov, chain, three, four, above, griffiths, simon, lieb, fkg, dimensional, with, transverse, field, renormalization, minimal, proof, comments, formula, for, spontaneous, magnetization,
Text of the page (most frequently used words):
the (754), model (197), displaystyle (176), and (168), ising (153), sigma (147), for (115), beta (102), with (99), that (88), spin (82), this (70), energy (62), are (59), sum (58), edit (49), doi (47), field (46), lattice (45), process (45), can (45), two (44), frac (42), #transition (41), phase (39), one (38), which (38), statistical (37), dimensional (37), bibcode (37), spins (35), cosh (35), from (34), temperature (34), was (33), critical (31), each (30), state (29), correlation (29), any (29), there (28), function (27), not (27), case (27), langle (27), rangle (27), all (26), probability (26), have (26), where (26), free (26), time (25), theory (25), system (25), sites (25), site (24), only (24), using (23), has (23), physics (23), left (23), right (23), models (22), mechanics (22), large (22), tree (22), magnetization (22), interaction (22), configuration (22), number (22), theorem (20), point (20), s2cid (20), between (20), isbn (19), when (19), zero (18), limit (18), 978 (18), neural (18), quantum (18), dimensions (18), given (18), sinh (18), hamiltonian (18), algorithm (18), random (17), external (17), solution (17), its (17), other (16), states (16), correlations (16), also (16), magnetic (16), since (16), graph (15), gamma (15), physical (15), cayley (15), onsager (15), 1007 (15), ferromagnetic (15), value (14), mathematical (14), arxiv (14), but (14), nearest (14), neighbor (14), same (14), markov (13), flip (13), 1103 (13), closed (13), exact (13), partition (13), equal (13), inequality (13), neighbors (13), finite (13), statistics (12), long (12), classical (12), average (12), article (12), three (12), behavior (12), review (12), conformal (12), new (12), functions (12), infinite (12), lambda (12), non (11), network (11), david (11), solved (11), peierls (11), main (11), following (11), range (11), cut (11), continuous (10), systems (10), independent (10), pmid (10), journal (10), issn (10), described (10), such (10), square (10), thermodynamic (10), first (10), small (10), sign (10), delta (10), exactly (9), formula (9), hopfield (9), branching (9), proof (9), single (9), dynamics (9), press (9), cambridge (9), renormalization (9), dimension (9), over (9), been (9), change (9), sqrt (9), set (9), toggle (8), volume (8), retrieved (8), machine (8), properties (8), length (8), simulation (8), monte (8), carlo (8), archived (8), order (8), original (8), 2015 (8), high (8), see (8), above (8), most (8), many (8), antiferromagnetic (8), cdots (8), positive (8), then (8), analytic (8), exponentially (8), ratio (8), possible (8), total (8), atoms (8), plus (8), wikipedia (7), terms (7), links (7), learning (7), equation (7), variables (7), paths (7), barth (7), yang (7), general (7), matrix (7), those (7), networks (7), boltzmann (7), than (7), argument (7), called (7), problem (7), used (7), shown (7), direction (7), into (7), boundary (7), metropolis (7), distribution (7), maximum (7), minus (7), size (7), about (6), text (6), under (6), may (6), cs1 (6), wayback (6), description (6), complete (6), extreme (6), doob (6), laws (6), constant (6), potts (6), local (6), interacting (6), chain (6), science (6), more (6), pdf (6), 2014 (6), glasser (6), spontaneous (6), university (6), lenz (6), their (6), 1999 (6), history (6), 1982 (6), line (6), transverse (6), methods (6), ferromagnetism (6), contribution (6), becomes (6), found (6), now (6), though (6), low (6), interactions (6), even (6), ldots (6), log (6), configurations (6), changes (6), position (6), depends (6), every (6), move (6), due (6), note (6), vertices (6), epsilon (6), adjacent (6), term (6), potential (6), pair (6), subsection (6), view (5), page (5), national (5), ergodic (5), martingale (5), tools (5), inequalities (5), self (5), lee (5), black (5), autoregressive (5), fields (5), variable (5), both (5), motion (5), transitions (5), well (5), simple (5), vol (5), related (5), 1016 (5), proceedings (5), 1963 (5), applications (5), sciences (5), bootstrap (5), exponents (5), how (5), group (5), results (5), pmc (5), ice (5), neurons (5), glass (5), 1972 (5), decay (5), result (5), goes (5), simplified (5), energies (5), without (5), exhibit (5), mechanical (5), does (5), amount (5), while (5), thermodynamics (5), geq (5), thus (5), represent (5), should (5), vertex (5), activity (5), neuron (5), gas (5), century (5), permanent (5), droplet (5), droplets (5), contents (4), search (4), available (4), commons (4), link (4), maint (4), fast (4), series (4), analysis (4), stochastic (4), space (4), stopping (4), representation (4), itô (4), white (4), percolation (4), gibbs (4), memory (4), additive (4), picture (4), example (4), 2020 (4), world (4), barry (4), 032141 (4), physreve (4), 1983 (4), 0378 (4), 4371 (4), physica (4), krizan (4), 1979 (4), physrev (4), planar (4), lars (4), 1944 (4), matter (4), phys (4), thermal (4), 1981 (4), simmons (4), duffin (4), 157 (4), 2012 (4), transfer (4), calculate (4), binding (4), sea (4), brain (4), glauber (4), correlated (4), information (4), simon (4), these (4), fluctuations (4), mean (4), proved (4), solutions (4), method (4), invariant (4), minimal (4), horizontal (4), expression (4), different (4), pairs (4), occurs (4), theta (4), cos (4), obtained (4), were (4), griffiths (4), obtain (4), prod (4), disordered (4), exp (4), never (4), lowest (4), having (4), factor (4), conditions (4), because (4), very (4), applied (4), sim (4), will (4), basic (4), acceptance (4), probabilities (4), must (4), within (4), equilibrium (4), often (4), further (4), questions (4), early (4), umezawa (4), account (4), topologies (4), topology (4), determined (4), trees (4), being (4), neighboring (4), bonds (4), greater (4), either (4), study (4), his (4), glasses (4), bits (4), once (4), electron (4), they (4), kramers (4), wannier (4), able (4), what (4), namely (4), hide (4), sidebar (4), code (3), contact (3), agree (3), wikimedia (3), inc (3), use (3), last (3), wikidata (3), list (3), articles (3), solvable (3), queueing (3), wiener (3), integral (3), skorokhod (3), reflection (3), principle (3), lévy (3), kolmogorov (3), exponential (3), convergence (3), maximal (3), logarithm (3), numbers (3), similar (3), cox (3), moving (3), integrated (3), boolean (3), measure (3), dynamical (3), jump (3), diffusion (3), geometric (3), dyson (3), discrete (3), magnets (3), heisenberg (3), larger (3), java (3), applet (3), online (3), ergodicity (3), 117 (3), american (3), 1952 (3), istrail (3), 2000 (3), second (3), discussion (3), citation (3), montroll (3), 1063 (3), ward (3), 2001 (3), scientific (3), 1989 (3), 1980 (3), 521 (3), nets (3), thesis (3), references (3), kadanoff (3), understanding (3), showk (3), sheer (3), paulos (3), miguel (3), poland (3), rychkov (3), slava (3), vichi (3), alessandro (3), solving (3), precise (3), 2016 (3), symmetry (3), 2013 (3), suggested (3), ichi (3), existence (3), ferromagnet (3), formalism (3), dna (3), 1073 (3), pnas (3), academy (3), like (3), sherrington (3), kirkpatrick (3), little (3), pattern (3), nakano (3), kaoru (3), associative (3), roy (3), behaviour (3), berry (3), pairwise (3), jaynes (3), 108 (3), 106 (3), 1432 (3), cluster (3), ernst (3), varying (3), still (3), four (3), cases (3), higher (3), rigorously (3), believed (3), theoretical (3), arguments (3), although (3), open (3), establish (3), physicists (3), active (3), structure (3), studied (3), translation (3), analytical (3), until (3), vertical (3), negative (3), specifically (3), make (3), gives (3), expansion (3), matrices (3), create (3), weighted (3), weight (3), domains (3), domain (3), step (3), magnetized (3), periodic (3), seen (3), begin (3), end (3), therefore (3), representing (3), strength (3), inverse (3), remains (3), includes (3), alpha (3), enough (3), present (3), furthermore (3), selection (3), otherwise (3), way (3), another (3), detailed (3), down (3), interest (3), numerical (3), significant (3), intermediate (3), phenomenon (3), cited (3), level (3), particles (3), vanishing (3), relevant (3), type (3), branches (3), joined (3), fact (3), increases (3), considered (3), established (3), increasing (3), generations (3), equiv (3), binary (3), published (3), paper (3), adding (3), weights (3), describe (3), typical (3), much (3), statistically (3), tends (3), compared (3), means (3), cost (3), atom (3), whether (3), magnetism (3), electrons (3), could (3), currents (3), atomic (3), after (3), show (3), duality (3), perimeter (3), box (3), temperatures (3), mostly (3), singularities (3), lead (3), did (3), maxwell (3), historical (3), significance (3), special (3), fkg (3), lieb (3), subset (3), aligned (3), edges (3), definition (3), einstein (3), languages (2), table (2), privacy (2), policy (2), additional (2), hidden (2), categories (2), category (2), deprecated (2), archival (2), service (2), url (2), multiple (2), names (2), authors (2), unsourced (2), statements (2), 2022 (2), short (2), problems (2), https (2), data (2), control (2), renewal (2), actuarial (2), mathematics (2), abstract (2), usual (2), snell (2), quadratic (2), prokhorov (2), optional (2), infinitesimal (2), meyer (2), decomposition (2), martin (2), law (2), weak (2), strong (2), theorems (2), central (2), feller (2), generalized (2), ross (2), variance (2), chen (2), conditional (2), heteroskedasticity (2), poisson (2), gaussian (2), differences (2), particle (2), walk (2), brownian (2), birth (2), entropy (2), biased (2), distributed (2), bernoulli (2), quanta (2), magazine (2), cartoon (2), transformed (2), provide (2), introduction (2), interactive (2), requires (2), impossible (2), sandia (2), lattices (2), export (2), siam (2), news (2), cipra (2), net (2), süzen (2), mehmet (2), effective (2), 25314429 (2), 2014phrve (2), 90c2141s (2), 1405 (2), 4497 (2), goldberg (2), jellito (2), 1970 (2), 1119 (2), 808 (2), sorin (2), acm (2), dimensionality (2), extended (2), john (2), 2007 (2), 1949 (2), 2009 (2), 308 (2), mccoy (2), 1920 (2), paperback (2), 981 (2), singapore (2), kleinert (2), lines (2), defects (2), 2008 (2), gauge (2), condensed (2), society (2), 2017 (2), 107 (2), itzykson (2), claude (2), drouffe (2), jean (2), michel (2), des (2), 1925 (2), springer (2), gallavotti (2), baierlein (2), 1967 (2), 883 (2), modern (2), encyclopedia (2), binder (2), baxter (2), rodney (2), london (2), academic (2), harcourt (2), brace (2), jovanovich (2), publishers (2), 0690578 (2), 083180 (2), cooperativity (2), leo (2), july (2), achieved (2), 174 (2), 2015jhep (2), 1502 (2), minimization (2), 914 (2), 119627708 (2), s10955 (2), 014 (2), 1042 (2), 2014jsp (2), 869e (2), 1403 (2), 4545 (2), 869 (2), 025022 (2), invariance (2), 012144 (2), operator (2), sphere (2), 021009 (2), inside (2), 055 (2), wilson (2), solve (2), june (2), 1978 (2), 642 (2), phases (2), fröhlich (2), spencer (2), 101 (2), comm (2), math (2), 1969 (2), vladimir (2), melt (2), 6056 (2), 1984 (2), 3088 (2), response (2), collective (2), computational (2), 2554 (2), 1975 (2), 1792 (2), 1974 (2), amari (2), shun (2), ieee (2), transactions (2), 1971 (2), 4615 (2), dependent (2), wang (2), dotsenko (2), dilute (2), 164 (2), schneidman (2), michael (2), segev (2), william (2), bialek (2), 440 (2), nature (2), imply (2), 1957 (2), 171 (2), 1957phrv (2), 620 (2), 113 (2), branch (2), richard (2), flagellar (2), switch (2), 327 (2), cooperative (2), 6399 (2), vincent (2), duminil (2), copin (2), hugo (2), sharpness (2), 0916 (2), 343 (2), communications (2), 1936 (2), application (2), combinatorial (2), optimization (2), 493 (2), stuart (2), computation (2), footnotes (2), vector (2), frustration (2), construction (2), parameter (2), defined (2), include (2), continuum (2), approximation (2), beyond (2), assuming (2), couplings (2), tells (2), known (2), fixed (2), simulations (2), describing (2), investigation (2), coupling (2), would (2), analogous (2), fermionic (2), derivation (2), 1951 (2), determinants (2), szegő (2), announced (2), partite (2), notably (2), around (2), antiparallel (2), cannot (2), 269185 (2), 2j_ (2), kt_ (2), calculated (2), derivative (2), int (2), proves (2), longer (2), computed (2), numerically (2), evaluate (2), functional (2), let (2), taking (2), textstyle (2), undergoes (2), upon (2), rewriting (2), basis (2), ordered (2), pauli (2), respective (2), depending (2), divided (2), density (2), configs (2), factorizes (2), tosses (2), reduces (2), proportional (2), off (2), reduced (2), difference (2), finding (2), extra (2), comments (2), choices (2), symmetric (2), eigenvalue (2), bmatrix (2), decays (2), lim (2), parameters (2), truncated (2), scaling (2), satisfied (2), hierarchical (2), comparison (2), exists (2), immediate (2), actually (2), version (2), flipping (2), opposite (2), form (2), too (2), eventually (2), pick (2), repeat (2), keep (2), less (2), flipped (2), follows (2), our (2), balance (2), selected (2), met (2), fluctuates (2), motivation (2), get (2), assume (2), hastings (2), near (2), commonly (2), chooses (2), out (2), accepted (2), stay (2), some (2), heat (2), magnet (2), absence (2), individual (2), consider (2), 500 (2), starting (2), including (2), interrelationships (2), underlying (2), natural (2), common (2), among (2), aspects (2), chemistry (2), itself (2), via (2), fundamental (2), addition (2), per (2), ground (2), possibility (2), divergent (2), convergent (2), based (2), exhibiting (2), relevance (2), biological (2), tend (2), firing (2), direct (2), top (2), counted (2), methodologies (2), valued (2), derived (2), section (2), equivalent (2), arbitrary (2), forming (2), indicates (2), exhibits (2), unusual (2), tau (2), tanh (2), investigate (2), arbitrarily (2), modelled (2), highly (2), activation (2), who (2), artificial (2), bond (2), leading (2), approach (2), later (2), reproduce (2), introduces (2), fire (2), along (2), lagrange (2), multipliers (2), sometimes (2), inactive (2), neuroscience (2), multiplicative (2), reinterpreted (2), changed (2), chemical (2), bit (2), occupied (2), kinetic (2), positions (2), clear (2), oriented (2), ask (2), side (2), interact (2), influence (2), 19th (2), thought (2), moments (2), stays (2), zeros (2), overall (2), separated (2), allowing (2), bounded (2), nonzero (2), establishes (2), quantitative (2), excess (2), snow (2), rudolf (2), correctly (2), everywhere (2), twentieth (2), formulated (2), atomism (2), real (2), gases (2), proven (2), positively (2), leq (2), respect (2), gave (2), 1924 (2), interacts (2), minimizing (2), edge (2), max (2), denotes (2), here (2), depended (2), subsets (2), complementary (2), undirected (2), connection (2), simplification (2), simplifications (2), desires (2), convention (2), moment (2), usually (2), wilhelm (2), landau (2), fermi (2), debye (2), dirac (2), bose (2), helmholtz (2), grand (2), isobaric (2), canonical (2), appearance (2), upload (2), file (2), read (2), donate (2), menu (2), add, topic, mobile, cookie, statement, developers, conduct, legal, safety, contacts, disclaimers, apply, you, registered, trademark, profit, organization, foundation, creative, attribution, sharealike, license, rendered, parsoid, edited, 2026, utc, webarchive, template, unfit, november, matches, org, index, php, title, ising_model, oldid, 1352146297, yale, lux, idref, israel, bnf, france, united, gnd, international, authority, databases, topics, signal, processing, ruin, finance, deviations, evt, econometrics, disciplines, hypotheses, uniform, integrability, stratonovich, hitting, tanaka, differential, envelope, variation, malliavin, calculus, metric, kosambi, karhunen, loève, extension, continuity, lemma, generator, girsanov, filtration, feynman, kac, dynkin, doléans, dade, cameron, marcinkiewicz, zygmund, kunita, watanabe, upcrossing, burkholder, davis, gundy, hewitt, savage, engelbert, schmidt, borel, cantelli, blumenthal, sanov, iterated, deviation, fisher, tippett, gnedenko, donsker, reversible, stationary, progressively, measurable, predictable, piecewise, deterministic, mixing, gauss, exchangeable, càdlàg, fluid, bulk, sparre, anderson, risk, cramér, lundberg, bühlmann, wilkie, vašíček, sabr, volatility, rendleman, bartter, libor, market, korn, kreer, lenssen, hull, heston, heath, jarrow, morton, hjm, garman, kohlhagen, ingersoll, cir, elasticity, cev, chan, karolyi, longstaff, sanders, ckls, scholes, karasinski, derman, toy, binomial, options, pricing, financial, garch, arma, arima, arch, determinantal, pitman, yor, dirichlet, noise, chains, regenerative, super, sub, hmm, sausage, telegraph, superprocess, stable, semimartingale, schramm, loewner, evolution, quasimartingale, homogeneous, compound, ornstein, uhlenbeck, mckean, vlasov, hunt, hawkes, fleming, viot, empirical, cauchy, meander, fractional, excursion, bridge, pure, death, bessel, airy, avoiding, loop, erased, moran, identically, galton, watson, chinese, restaurant, processes, julia, github, com, cossio, squareisingmodel, good, tong, lecture, notes, image, denoising, graphics, webgl, compatible, browser, researcher, claims, enrique, zeleny, wolfram, demonstrations, project, dirk, brockman, allows, users, working, presentation, slide, sing, tempered, complicated, ucsc, edition, arthur, advance, media, 672, 90138, 1983phya, 670g, 670, 117a, north, holland, publishing, 242, 90157, 230, 119a, 280, 90134, 1979phya, 268j, 268, 99a, 1036, 1976530, 1970amjph, 1033g, 1033, 816, 0051740, 1952phrv, 808y, 7944336, 2114521, 58113, 184, 1145, 335305, 335316, thirty, annual, symposium, computing, completeness, universality, intractability, across, surfaces, palmer, birkhäuser, boston, 8176, 4248, 261, supplemento, nuovo, cimento, crystal, disorder, 149, 0010315, 1944phrv, 117o, elliott, renfrey, 322, 0148406, 0022, 2488, 1703955, 1963jmp, 308m, tai, tsun, 1973, harvard, massachusetts, 674, 91440, beiträge, zum, verständnis, der, magnetischen, eigenschaften, festen, körpern, 615, 613, physikalische, zeitschrift, schulte, frohlinde, 2011, 4658, theories, superflow, vortex, 742, stresses, 743, 1456, 9971, 0210, kindermann, laurie, 8218, 3381, friedli, velenik, 18482, concrete, 40805, savoirs, actuels, edp, editions, 86883, 360, cnrs, théorie, statistique, champs, beitrag, zur, theorie, ferromagnetismus, 258, 122157319, bf02980577, 1925zphy, 253i, 253, 1987, 2nd, wiley, 471, 81518, huang, kerson, texts, monographs, berlin, verlag, 1707309, 540, 64883, 662, 03952, 59082, brush, stephen, 893, revmodphys, 1967rvmp, 883b, reviews, 1994, ems, burlington, vermont, 118, 8231704, oclc, master, april, club, deep, semidefinite, program, solver, 35625559, 1029, 8479, jhep06, 174s, 02033, 39692193, physrevd, 2012phrvd, 86b5022e, 1203, 6064, d86, rev, delamotte, bertrand, tissier, matthieu, wschebor, nicolás, scale, implies, 12144, 14538564, 26871060, 2016phrve, 93a2144d, 1501, 01776, zhu, wei, han, chao, huffman, emilie, hofmann, johannes, yin, 2023, uncovering, correspondence, fuzzy, regularization, 253107625, physrevx, 2023phrvx, 13b1009z, 2210, 13482, cosme, catarina, lopes, viana, parente, penedones, joao, 53710971, jhep08, 022, 022c, 1503, 02011, billó, caselle, gaiotto, gliozzi, meineri, others, 119226610, jhep07, 2013jhep, 055b, 1304, 4110, 1307, jhep, ken, recalls, murray, gell, mann, wood, charlie, maris, humphrey, 657, 0002, 9505, 11224, 1978amjph, 652m, 652, teaching, suzuki, sei, inoue, jun, chakrabarti, bikas, 33038, 33039, 122722140, bf01208373, 1982cmaph, 87f, 122117175, bf01645907, 1969cmaph, 91d, ruelle, 4495, rigorous, teif, e80, 17526526, 1920246, 1093, nar, gkm268, nucleic, acids, res, protein, drug, gene, regulation, september, 118355454, suzen, newman, barkema, clarendon, 851797, ping, ivan, sudakov, courtenay, kenneth, golden, ponds, arctic, 1408, 2487v3, seung, sompolinsky, tishby, 1992, 6091, 9907706, physreva, 1992phrva, 6056s, examples, engel, broeck, van, den, york, 77307, 3092, 6587342, 345226, 1984pnas, 3088h, graded, 2558, 6953413, 346238, 1982pnas, 2554h, emergent, abilities, scott, 1796, 0031, 9007, physrevlett, 1975phrvl, 1792s, letters, persistent, 120, 0025, 5564, 90031, biosciences, patterns, sequences, organizing, threshold, elements, 1206, 1197, associatron, smc, 388, 1109, tsmc, 4309133, 1972itsmc, 380n, 380, man, cybernetics, 186, 7568, 7566, 5_15, 172, recognition, february, 307, 2021, 1703954, 294, andreichenko, 1990, 239, 90196, 1990phya, 221w, 221, selke, elad, ronen, 2006, 7087, 1012, 16625187, 1785327, 1038, nature04701, 2006natur, 1007s, bio, 0512013, strongly, population, edwin, 190, 171j, 630, 17870175, 620j, vtyurina, natalia, dulin, docter, margreet, anne, dekker, nynke, abbondanzieri, elio, 27091987, 4983820, 0027, 8424, 1521241113, 2016pnas, 4982v, 4982, hysteresis, compaction, dps, bai, fan, nicolau, dan, pilizota, teuta, steel, bradley, maini, philip, 2010, conformational, spread, mechanism, bacterial, 5966, 689, 206523521, 20133571, 0036, 8075, 1126, 1182105, 2010sci, 685b, 685, shi, duke, 1998, bacteril, sensing, 6406, 18854281, 1998phrve, 6399s, 9901052, beffara, 542, 55391558, 2064, s00440, 011, 0353, 511, 153, dual, tassion, 745, 119330137, s00220, 015, 2480, 2016cmaph, 725d, 03050, 725, ferromagnets, 126, 17543488, bf01982711, 1980cmaph, 111s, 111, 309, 481, 122630492, 1017, s0305004100019174, 1936pcps, 477p, 477, philosophical, barahona, francisco, jünger, reinelt, gerhard, 1988, circuit, layout, design, 513, 170992, jstor, 0030, 364x, 1287, opre, operations, research, grötschel, chapter, samuel, anticommuting, integrals, 2814, 524404, 1980jmp, 2806s, 2806, chapters, vii, evenness, kuramoto, rotor, geometrical, irreducible, annni, productively, locally, region, entire, slow, variations, averaging, moves, accuracy, improves, deeper, behaves, going, approximations, intractable, concerns, spatially, nothing, nonplanar, laboratories, shows, sense, diverges, transformation, unitary, evidenced, diagonalization, compute, experiments, currently, yields, cubic, theoreticians, searched, decades, exist, strings, carried, conjecturally, unknown
Text of the page (random words):
starting with all the spins minus to fluctuate to a state where most of the spins are plus for this to happen droplets of plus spin must be able to congeal to make the plus state the energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet l where plus spins and minus spins neighbor each other for a droplet with perimeter l the area is somewhere between l 2 2 the straight line and l 4 2 the square box the probability cost for introducing a droplet has the factor e β l but this contributes to the partition function multiplied by the total number of droplets with perimeter l which is less than the total number of paths of length l n l 4 2 l displaystyle n l 4 2l so that the total spin contribution from droplets even overcounting by allowing each site to have a separate droplet is bounded above by l l 2 4 2 l e 4 β l displaystyle sum _ l l 2 4 2l e 4 beta l which goes to zero at large β for β sufficiently large this exponentially suppresses long loops so that they cannot occur and the magnetization never fluctuates too far from 1 so peierls established that the magnetization in the ising model eventually defines superselection sectors separated domains not linked by finite fluctuations kramers wannier duality edit main article kramers wannier duality kramers and wannier were able to show that the high temperature expansion and the low temperature expansion of the model are equal up to an overall rescaling of the free energy this allowed the phase transition point in the two dimensional model to be determined exactly under the assumption that there is a unique critical point yang lee zeros edit main article lee yang theorem after onsager s solution yang and lee investigated the way in which the partition function becomes singular as the temperature approaches the critical temperature applications edit magnetism edit the original motivation for the model was the phenomenon of ferromagnetism iron is magnetic once it is magnetized it stays magnetized for a long time compared to any atomic time in the 19th century it was thought that magnetic fields are due to currents in matter and ampère postulated that permanent magnets are caused by permanent atomic currents the motion of classical charged particles could not explain permanent currents though as shown by larmor in order to have ferromagnetism the atoms must have permanent magnetic moments which are not due to the motion of classical charges once the electron s spin was discovered it was clear that the magnetism should be due to a large number of electron spins all oriented in the same direction it was natural to ask how the electrons spins all know which direction to point in because the electrons on one side of a magnet don t directly interact with the electrons on the other side they can only influence their neighbors the ising model was designed to investigate whether a large fraction of the electron spins could be oriented in the same direction using only local forces lattice gas edit the ising model can be reinterpreted as a statistical model for the motion of atoms since the kinetic energy depends only on momentum and not on position while the statistics of the positions only depends on the potential energy the thermodynamics of the gas only depends on the potential energy for each configuration of atoms a coarse model is to make space time a lattice and imagine that each position either contains an atom or it doesn t the space of configuration is that of independent bits b i where each bit is either 0 or 1 depending on whether the position is occupied or not an attractive interaction reduces the energy of two nearby atoms if the attraction is only between nearest neighbors the energy is reduced by 4 jb i b j for each occupied neighboring pair the density of the atoms can be controlled by adding a chemical potential which is a multiplicative probability cost for adding one more atom a multiplicative factor in probability can be reinterpreted as an additive term in the logarithm the energy the extra energy of a configuration with n atoms is changed by μn the probability cost of one more atom is a factor of exp βμ so the energy of the lattice gas is e 1 2 i j 4 j b i b j i μ b i displaystyle e frac 1 2 sum _ langle i j rangle 4jb_ i b_ j sum _ i mu b_ i rewriting the bits in terms of spins b i s i 1 2 displaystyle b_ i s_ i 1 2 e 1 2 i j j s i s j 1 2 i 4 j μ s i displaystyle e frac 1 2 sum _ langle i j rangle js_ i s_ j frac 1 2 sum _ i 4j mu s_ i for lattices where every site has an equal number of neighbors this is the ising model with a magnetic field h zj μ 2 where z is the number of neighbors in biological systems modified versions of the lattice gas model have been used to understand a range of binding behaviors these include the binding of ligands to receptors in the cell surface 12 the binding of chemotaxis proteins to the flagellar motor 13 and the condensation of dna 14 neuroscience edit the activity of neurons in the brain can be modelled statistically each neuron at any time is either active or inactive the active neurons are those that send an action potential down the axon in any given time window and the inactive ones are those that do not following the general approach of jaynes 15 16 a later interpretation of schneidman berry segev and bialek 17 is that the ising model is useful for any model of neural function because a statistical model for neural activity should be chosen using the principle of maximum entropy given a collection of neurons a statistical model which can reproduce the average firing rate for each neuron introduces a lagrange multiplier for each neuron e i h i s i displaystyle e sum _ i h_ i s_ i but the activity of each neuron in this model is statistically independent to allow for pair correlations when one neuron tends to fire or not to fire along with another introduce pair wise lagrange multipliers e 1 2 i j j i j s i s j i h i s i displaystyle e tfrac 1 2 sum _ ij j_ ij s_ i s_ j sum _ i h_ i s_ i where j i j displaystyle j_ ij are not restricted to neighbors note that this generalization of ising model is sometimes called the quadratic exponential binary distribution in statistics this energy function only introduces probability biases for a spin having a value and for a pair of spins having the same value higher order correlations are unconstrained by the multipliers an activity pattern sampled from this distribution requires the largest number of bits to store in a computer in the most efficient coding scheme imaginable as compared with any other distribution with the same average activity and pairwise correlations this means that ising models are relevant to any system which is described by bits which are as random as possible with constraints on the pairwise correlations and the average number of 1s which frequently occurs in both the physical and social sciences spin glasses edit with the ising model the so called spin glasses can also be described by the usual hamiltonian h 1 2 j i k s i s k textstyle h frac 1 2 sum j_ i k s_ i s_ k where the s variables describe the ising spins while the j i k are taken from a random distribution for spin glasses a typical distribution chooses antiferromagnetic bonds with probability p and ferromagnetic bonds with probability 1 p also known as the random bond ising model these bonds stay fixed or quenched even in the presence of thermal fluctuations when p 0 we have the original ising model this system deserves interest in its own particularly one has non ergodic properties leading to strange relaxation behaviour much attention has been also attracted by the related bond and site dilute ising model especially in two dimensions leading to intriguing critical behavior 18 artificial neural network edit main article hopfield network the ising model was instrumental in the development of the hopfield network the original ising model is a model for equilibrium roy j glauber in 1963 studied the ising model evolving in time as a process towards thermal equilibrium glauber dynamics adding in the component of time 19 kaoru nakano 1971 20 21 and shun ichi amari 1972 22 proposed to modify the weights of an ising model by hebbian learning rule as a model of associative memory the same idea was published by william a little de 1974 23 who was cited by hopfield in his 1982 paper the sherrington kirkpatrick model of spin glass published in 1975 24 is the hopfield network with random initialization sherrington and kirkpatrick found that it is highly likely for the energy function of the sk model to have many local minima in the 1982 paper hopfield applied this recently developed theory to study the hopfield network with binary activation functions 25 in a 1984 paper he extended this to continuous activation functions 26 it became a standard model for the study of neural networks through statistical mechanics 27 28 sea ice edit the melt pond can be modelled by the ising model sea ice topography data bears rather heavily on the results the state variable is binary for a simple 2d approximation being either water or ice 29 cayley tree topologies and large neural networks edit an open cayley tree or branch with branching ratio 2 and k generations in order to investigate an ising model with potential relevance for large e g with 10 4 displaystyle 10 4 or 10 5 displaystyle 10 5 interactions per node neural nets at the suggestion of krizan in 1979 barth 1981 obtained the exact analytical expression for the free energy of the ising model on the closed cayley tree with an arbitrarily large branching ratio for a zero external magnetic field in the thermodynamic limit by applying the methodologies of glasser 1970 and jellito 1979 β f ln 2 2 γ γ 1 ln cosh j γ γ 1 γ 1 i 2 z 1 γ i ln j i τ displaystyle beta f ln 2 frac 2 gamma gamma 1 ln cosh j frac gamma gamma 1 gamma 1 sum _ i 2 z frac 1 gamma i ln j_ i tau a closed cayley tree with branching ratio 4 only sites for generations k k 1 and k 1 overlapping as one row are shown for the joined trees where γ displaystyle gamma is an arbitrary branching ratio greater than or equal to 2 t tanh j displaystyle t equiv tanh j τ t 2 displaystyle tau equiv t 2 j β ϵ displaystyle j equiv beta epsilon with ϵ displaystyle epsilon representing the nearest neighbor interaction energy and there are k in the thermodynamic limit generations in each of the tree branches forming the closed tree architecture as shown in the given closed cayley tree diagram the sum in the last term can be shown to converge uniformly and rapidly i e for z it remains finite yielding a continuous and monotonous function establishing that for γ displaystyle gamma greater than or equal to 2 the free energy is a continuous function of temperature t further analysis of the free energy indicates that it exhibits an unusual discontinuous first derivative at the critical temperature krizan barth glasser 1983 glasser goldberg 1983 the spin spin correlation between sites in general m and n on the tree was found to have a transition point when considered at the vertices e g a and ā its reflection their respective neighboring sites such as b and its reflection and between sites adjacent to the top and bottom extreme vertices of the two trees e g a and b as may be determined from s m s n z n 1 0 t cosh j n b 2 n l 1 z g m n l t l displaystyle langle s_ m s_ n rangle z_ n 1 0 t cosh j n_ b 2 n sum _ l 1 z g_ mn l t l where n b displaystyle n_ b is equal to the number of bonds g m n l t l displaystyle g_ mn l t l is the number of graphs counted for odd vertices with even intermediate sites see cited methodologies and references for detailed calculations 2 n displaystyle 2 n is the multiplicity resulting from two valued spin possibilities and the partition function z n displaystyle z_ n is derived from s e β h displaystyle sum _ s e beta h note s i displaystyle s_ i is consistent with the referenced literature in this section and is equivalent to s i displaystyle s_ i or σ i displaystyle sigma _ i utilized above and in earlier sections it is valued at 1 displaystyle pm 1 the critical temperature t c displaystyle t_ c is given by t c 2 ϵ k b ln γ 1 ln γ 1 displaystyle t_ c frac 2 epsilon k_ text b ln sqrt gamma 1 ln sqrt gamma 1 the critical temperature for this model is only determined by the branching ratio γ displaystyle gamma and the site to site interaction energy ϵ displaystyle epsilon a fact which may have direct implications associated with neural structure vs its function in that it relates the energies of interaction and branching ratio to its transitional behavior for example a relationship between the transition behavior of activities of neural networks between sleeping and wakeful states which may correlate with a spin spin type of phase transition in terms of changes in neural interconnectivity γ displaystyle gamma and or neighbor to neighbor interactions ϵ displaystyle epsilon over time is just one possible avenue suggested for further experimental investigation into such a phenomenon in any case for this ising model it was established that the stability of the long range correlation increases with increasing γ displaystyle gamma or increasing ϵ displaystyle epsilon for this topology the spin spin correlation was found to be zero between the extreme vertices and the central sites at which the two trees or branches are joined i e between a and individually c d or e this behavior is explained to be due to the fact that as k increases the number of links increases exponentially between the extreme vertices and so even though the contribution to spin correlations decrease exponentially the correlation between sites such as the extreme vertex a in one tree and the extreme vertex in the joined tree ā remains finite above the critical temperature in addition a and b also exhibit a non vanishing correlation as do their reflections thus lending itself to for b level sites with a level being considered clusters which tend to exhibit synchronization of firing based upon a review of other classical network models as a comparison the ising model on a closed cayley tree was determined to be the first classical statistical mechanical model to demonstrate both local and long range sites with non vanishing spin spin correlations while at the same time exhibiting intermediate sites with zero correlation which indeed was a relevant matter for large neural networks at the time of its consideration the model s behavior is also of relevance for any other divergent convergent tree physical or biological system exhibiting a closed cayley tree topology with an ising type of interaction this topology should not be ignored since its behavior for ising models has been solved exactly and presumably nature will have found a way of taking advantage of such simple symmetries at many levels of its designs barth 1981 early on noted the possibility of interrelationships between 1 the classic...
|
