. "Die Fokker-Planck-Gleichung (FPG, nach Adriaan Dani\u00EBl Fokker (1887\u20131972) und Max Planck (1858\u20131947)) ist eine partielle Differentialgleichung. Sie beschreibt die zeitliche Entwicklung einer Wahrscheinlichkeitsdichtefunktion unter der Wirkung von Drift und Diffusion . In ihrer eindimensionalen Form lautet die Gleichung: F\u00FCr verschwindende Drift und konstante Diffusion geht die FPG in die Diffusions- (oder auch W\u00E4rmeleitungs-) Gleichung \u00FCber. In Dimensionen lautet die Fokker-Planck-Gleichung Von der Smoluchowski-Gleichung spricht man, wenn die Positionen der Teilchen im System beschreibt."@de . . . . . "\u0420\u0456\u0432\u043D\u044F\u0301\u043D\u043D\u044F \u0424\u043E\u0301\u043A\u043A\u0435\u0440\u0430 \u2014 \u041F\u043B\u0430\u0301\u043D\u043A\u0430 \u2014 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0432 \u0447\u0430\u0441\u0442\u0438\u043D\u043D\u0438\u0445 \u043F\u043E\u0445\u0456\u0434\u043D\u0438\u0445, \u0449\u043E \u043E\u043F\u0438\u0441\u0443\u0454 \u0435\u0432\u043E\u043B\u044E\u0446\u0456\u044E \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0432\u0438\u043F\u0430\u0434\u043A\u043E\u0432\u043E\u0457 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0438. \u0414\u043B\u044F \u043E\u0434\u043D\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0432\u0438\u043F\u0430\u0434\u043A\u043E\u0432\u043E\u0457 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0438 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0424\u043E\u043A\u043A\u0435\u0440\u0430-\u041F\u043B\u0430\u043D\u043A\u0430 \u043C\u0430\u0454 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0438\u0439 \u0432\u0438\u0433\u043B\u044F\u0434 , \u0434\u0435 \u2014 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0432\u0438\u043F\u0430\u0434\u043A\u043E\u0432\u043E\u0457 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0438, \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0434\u0440\u0435\u0439\u0444\u043E\u0432\u0438\u043C \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u043E\u043C, \u0430 \u2014 \u0434\u0438\u0444\u0443\u0437\u0456\u0439\u043D\u0438\u043C \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u043E\u043C. \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u0443 \u0432\u0438\u043F\u0430\u0434\u043A\u0443 \u0431\u0440\u043E\u0443\u043D\u0456\u0432\u0441\u044C\u043A\u043E\u0433\u043E \u0440\u0443\u0445\u0443 \u0432\u0437\u0434\u043E\u0432\u0436 \u043F\u0440\u044F\u043C\u043E\u0457 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0424\u043E\u043A\u043A\u0435\u0440\u0430-\u041F\u043B\u0430\u043D\u043A\u0430 \u0434\u043B\u044F \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0447\u0430\u0441\u0442\u0438\u043D\u043E\u043A \u0437\u0430 \u0448\u0432\u0438\u0434\u043A\u043E\u0441\u0442\u044F\u043C\u0438 \u043C\u0430\u0454 \u0432\u0438\u0433\u043B\u044F\u0434: , \u0434\u0435 \u2014 \u0448\u0432\u0438\u0434\u043A\u0456\u0441\u0442\u044C \u0431\u0440\u043E\u0443\u043D\u0456\u0432\u0441\u044C\u043A\u043E\u0457 \u0447\u0430\u0441\u0442\u043A\u0438, \u2014 \u0457\u0457 \u043C\u0430\u0441\u0430, \u2014 \u0441\u0442\u0430\u043B\u0430 \u0411\u043E\u043B\u044C\u0446\u043C\u0430\u043D\u0430, T \u2014 \u0442\u0435\u043C\u043F\u0435\u0440\u0430\u0442\u0443\u0440\u0430, \u2014 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442 \u0432'\u044F\u0437\u043A\u043E\u0441\u0442\u0456, \u0440\u043E\u0437\u0434\u0456\u043B\u0435\u043D\u0438\u0439 \u043D\u0430 \u043C\u0430\u0441\u0443 \u0447\u0430\u0441\u0442\u043A\u0438. \u0414\u0438\u0444\u0443\u0437\u0456\u0439\u043D\u0438\u0439 \u0456 \u0434\u0440\u0435\u0439\u0444\u043E\u0432\u0438\u0439 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0438 \u043C\u043E\u0436\u043D\u0430 \u043E\u0442\u0440\u0438\u043C\u0430\u0442\u0438, \u0440\u043E\u0437\u0433\u043B\u044F\u0434\u0430\u044E\u0447\u0438 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u041B\u0430\u043D\u0436\u0435\u0432\u0435\u043D\u0430."@uk . . . . . . . "In matematica e nella teoria della probabilit\u00E0, l'equazione di Fokker-Planck, il cui nome \u00E8 dovuto a Adriaan Fokker e a Max Planck, detta anche equazione anticipativa di Kolmogorov, descrive l'evoluzione temporale della funzione di densit\u00E0 di probabilit\u00E0 della posizione di una particella, e pu\u00F2 essere generalizzata ad altri enti osservabili. Il primo impiego dell'equazione di Fokker-Planck fu la descrizione statistica del moto browniano di una particella in un fluido. In una dimensione spaziale , l'equazione di Fokker-Planck per un processo con termine di deriva e termine di diffusione \u00E8: Pi\u00F9 in generale, la probabilit\u00E0 tempo-dipendente della distribuzione potrebbe dipendere da un set di macrovariabili . La forma generale dell'equazione di Fokker-Planck \u00E8 quindi: dove \u00E8 il vettore di direzione e il tensore di diffusione, quest'ultimo dei quali risulta dalla presenza della forza stocastica."@it . . . . . "\u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631 \u0628\u0644\u0627\u0646\u0643 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Fokker-Planck equation)\u200F \u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0635\u0641 \u0627\u0644\u062A\u0637\u0648\u0631 \u0627\u0644\u0632\u0645\u0646\u064A \u00AB\u0627\u0644\u062A\u063A\u064A\u0631 \u062E\u0644\u0627\u0644 \u0627\u0644\u0632\u0645\u0646\u00BB \u0644\u062A\u0627\u0628\u0639 \u0627\u0644\u0643\u062B\u0627\u0641\u0629 \u0627\u0644\u0627\u062D\u062A\u0645\u0627\u0644\u064A\u0629 \u0644\u0633\u0631\u0639\u0629 \u062C\u0633\u064A\u0645 \u0645\u0627. \u0648\u064A\u0645\u0643\u0646 \u0627\u0646 \u062A\u0643\u062A\u0628 \u0628\u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u0639\u0627\u0645 \u0639\u0644\u0649 \u0634\u0643\u0644 \u0627\u0628\u0639\u0627\u062F \u0645\u062A\u0639\u062F\u062F\u0629 \u0628\u0627\u0633\u062A\u062E\u062F\u0627\u0645 \u0627\u0644\u0645\u0624\u062B\u0631 \u0646\u0628\u0644\u0627. \u0644\u0643\u0646\u0647\u0627 \u063A\u0627\u0644\u0628\u0627 \u0645\u0627 \u062A\u0643\u062A\u0628 \u0628\u0627\u062E\u062A\u0635\u0627\u0631 \u0639\u0644\u0649 \u0627\u0639\u062A\u0628\u0627\u0631 \u0627\u0644\u062D\u0631\u0643\u0629 \u0630\u0627\u062A \u0628\u0639\u062F \u0648\u0627\u062D\u062F. \u062A\u0639\u0648\u062F \u062A\u0633\u0645\u064A\u062A\u0647\u0627 \u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u0627\u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0623\u0644\u0645\u0627\u0646\u064A \u0627\u0644\u0634\u0647\u064A\u0631 \u0635\u0627\u062D\u0628 \u0648\u0625\u0644\u0649 \u0627\u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0647\u0648\u0644\u0646\u062F\u064A \u0627\u062F\u0631\u064A\u0627\u0646 \u0641\u0648\u0643\u0631 \u0648\u0642\u062F \u062A\u0633\u0645\u0649 \u0623\u062D\u064A\u0627\u0646\u0627 \u0628\u0623\u0633\u0645\u0627\u0621 \u0623\u062E\u0631\u0649 \u0645\u0646\u0647\u0627 \u0645\u0639\u0627\u062F\u0644\u0629 \u0643\u0648\u0644\u0648\u0643\u0644\u0648\u0641 \u0627\u0644\u0645\u062A\u0642\u062F\u0645\u0629 \"Kolmogorov Forward Equation\" \u0648\u0645\u0639\u0627\u062F\u0644\u0629 \u0633\u0645\u0648\u0644\u0648\u0643\u0648\u0641\u0633\u0643\u064A \"Smoluchowski equation\" \u0648\u062E\u0627\u0635 \u0639\u0646\u062F\u0645\u0627 \u062A\u0635\u0641 \u0647\u0630\u0647 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u062A\u0648\u0632\u0639\u0627\u062A \u0627\u0644\u0627\u062D\u062A\u0645\u0627\u0644\u064A\u0629 \u0627\u0644\u0645\u0645\u0643\u0646\u0629 \u0644\u0645\u0648\u0636\u0639 \u062C\u0633\u064A\u0645 \u0623\u064A \u062A\u063A\u064A\u0631 \u0627\u062D\u062A\u0645\u0627\u0644 \u0623\u0645\u0627\u0643\u0646 \u062A\u0648\u0627\u062C\u062F \u062C\u0633\u064A\u0645 \u0641\u064A \u0646\u0642\u0637\u0629 \u0645\u0639\u064A\u0646\u0629 \u062E\u0644\u0627\u0644 \u0627\u0644\u0632\u0645\u0646\" \u0648\u0642\u062F \u0623\u062C\u0631\u064A \u0623\u0648\u0644 \u0627\u0634\u062A\u0642\u0627\u0642 \u0645\u062C\u0647\u0631\u064A \u0645\u062A\u0633\u0642 \u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631 \u0628\u0644\u0627\u0646\u0643 \u0641\u064A \u0645\u062E\u0637\u0637 \u0648\u0627\u062D\u062F \u0645\u0646 \u0627\u0644\u0645\u064A\u0643\u0627\u0646\u064A\u0643\u0627 \u0627\u0644\u0643\u0644\u0627\u0633\u064A\u0643\u064A\u0629 \u0648\u0627\u0644\u0643\u0645\u064A\u0629 \u0628\u0648\u0627\u0633\u0637\u0629 \u0645\u0646 \u0642\u0628\u0644 \u0646\u064A\u0643\u0648\u0644\u0627\u064A \u0628\u0648\u063A\u0648\u0644\u064A\u0648\u0628\u0648\u0641 . \u0641\u064A \u0628\u0639\u062F \u0645\u0643\u0627\u0646\u064A \u0648\u0627\u062D\u062F x, \u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631 \u2013 \u0628\u0644\u0627\u0646\u0643 \u0644\u0645\u0639\u0644\u0627\u062C \u0645\u0639 \u0627\u0646\u062C\u0631\u0627\u0641 \u0627\u064A\u062A\u0648 D1(x,t) \u0648\u0646\u0634\u0631 D2(x,t) \u0647\u0648: \u0648\u0645\u0639 \u0630\u0644\u0643\u060C \u0641\u064A \u0643\u062B\u064A\u0631 \u0645\u0646 \u0627\u0644\u0623\u062D\u064A\u0627\u0646\u060C \u0641\u064A \u0627\u0644\u062A\u0637\u0628\u064A\u0642\u0627\u062A \u0627\u0644\u0641\u064A\u0632\u064A\u0627\u0626\u064A\u0629 \u0646\u0623\u062E\u0630 \u0641\u064A \u0627\u0644\u0627\u0639\u062A\u0628\u0627\u0631 \u0645\u0639\u0644\u0627\u062C \u0633\u062A\u0631\u0627\u062A\u0648\u0646\u0648\u0641\u064A\u062A\u0634 \u0630\u0648 \u0627\u0644\u0635\u0644\u0629 \u0623\u0643\u062B\u0631 (\u0645\u0643\u062A\u0648\u0628 \u0641\u064A \u0634\u0643\u0644 \u0627\u064A\u062A\u0648): \u0648\u0627\u0644\u0630\u064A \u064A\u062A\u0636\u0645\u0646 \u0639\u0627\u0645\u0644 \u0627\u0644\u0627\u0646\u062C\u0631\u0627\u0641 \u0627\u0644\u0645\u0636\u0627\u0641 \u0648\u0630\u0644\u0643 \u0628\u0633\u0628\u0628 \u0627\u0644\u0622\u062B\u0627\u0631 \u0627\u0644\u0627\u0646\u062A\u0634\u0627\u0631 \u0627\u0644\u0645\u062A\u062F\u0631\u062C.\u0623\u0643\u062B\u0631 \u0639\u0645\u0648\u0645\u0627\u060C \u0627\u0644\u062A\u0648\u0632\u064A\u0639 \u0627\u0644\u0627\u062D\u062A\u0645\u0627\u0644\u064A \u0627\u0644\u0645\u0639\u062A\u0645\u062F \u0639\u0644\u0649 \u0627\u0644\u0632\u0645\u0646 \u064A\u0639\u062A\u0645\u062F \u0639\u0644\u0649 \u0645\u062C\u0645\u0648\u0639\u0629 \u0645\u0646 \u0645\u062A\u063A\u064A\u0631\u0627\u062A \u0643\u0644\u064A\u0629 . \u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0639\u0627\u0645 \u0644\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631 \u0628\u0644\u0627\u0646\u0643 \u064A\u0635\u0628\u062D \u0628\u0639\u062F \u0630\u0644\u0643: \u062D\u064A\u062B \u0647\u0648 \u0645\u062A\u062C\u0647 \u0627\u0644\u0627\u0646\u062C\u0631\u0627\u0641 \u0648 \u0647\u0648 \u0645\u0648\u062A\u0631 \u0627\u0644\u0627\u0646\u062A\u0634\u0627\u0631; \u0648\u064A\u0646\u062A\u062C \u0647\u0630\u0647 \u0627\u0644\u0623\u062E\u064A\u0631 \u0645\u0646 \u0648\u062C\u0648\u062F \u0627\u0644\u0642\u0648\u0629 \u0627\u0644\u062A\u0635\u0627\u062F\u0641\u064A\u0629.."@ar . "A equa\u00E7\u00E3o de Fokker\u2013Planck, denominada assim por Adriaan Fokker e Max Planck, e tamb\u00E9m conhecida como equa\u00E7\u00E3o avan\u00E7ada de Kolmog\u00F3rov (por Andr\u00E9i Kolmog\u00F3rov, quem primeiro a introduziu em um artigo de 1931 ), descreve a evolu\u00E7\u00E3o temporal da fun\u00E7\u00E3o de densidade de probabilidade que mostra a posi\u00E7\u00E3o e a velocidade de uma part\u00EDcula, ainda que possa ser generalizada a outro tipo de vari\u00E1veis. A equa\u00E7\u00E3o \u00E9 aplic\u00E1vel a sistemas que possam ser descritos por um pequeno n\u00FAmero de \"macrovari\u00E1veis\", onde outros par\u00E2metros variam t\u00E3o rapidamente com o tempo que podem ser tratados como \"ru\u00EDdo\" ou uma perturba\u00E7\u00E3o."@pt . . . . . . . . . "\u0423\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0424\u043E\u043A\u043A\u0435\u0440\u0430 \u2014 \u041F\u043B\u0430\u043D\u043A\u0430"@ru . . . . . "\u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631\u2013\u0628\u0644\u0627\u0646\u0643"@ar . . . . . . . "\u798F\u514B-\u666E\u6717\u514B\u65B9\u7A0B\uFF08Fokker\u2013Planck equation\uFF09\u63CF\u8FF0\u7C92\u5B50\u5728\u4F4D\u80FD\u5834\u4E2D\u53D7\u5230\u96A8\u6A5F\u529B\u5F8C\uFF0C\u96A8\u6642\u9593\u6F14\u5316\u7684\u4F4D\u7F6E\u6216\u662F\u901F\u5EA6\u7684\u5206\u5E03\u51FD\u6578 \u3002\u6B64\u65B9\u7A0B\u5F0F\u4EE5\u8377\u862D\u7269\u7406\u5B78\u5BB6\u963F\u5FB7\u91CC\u5B89\u00B7\u798F\u514B\u8207\u99AC\u514B\u65AF\u00B7\u666E\u6717\u514B\u7684\u59D3\u6C0F\u4F86\u547D\u540D\u3002 \u4E00\u7DAD x\u65B9\u5411\u4E0A,\u798F\u514B-\u666E\u6717\u514B\u65B9\u7A0B\u6709\u5169\u500B\u53C3\u6578\uFF0C\u4E00\u662F\u62D6\u66F3\u53C3\u6578 D1(x,t)\uFF0C\u53E6\u4E00\u662F\u64F4\u6563 D2(x,t) \u5728 \u7DAD\u7A7A\u9593\u4E2D\u7684\u798F\u514B-\u666E\u6717\u514B\u65B9\u7A0B\u662F \u662F\u7B2C\u7DAD\u5EA6\u7684\u4F4D\u7F6E\uFF0C\u6B64\u6642 \u70BA\u62D6\u66F3\u5411\u91CF\uFF0C\u70BA\u64F4\u6563\u5F35\u91CF\u3002"@zh . . . "R\u00F3wnanie Fokkera-Plancka"@pl . "L'\u00E9quation de Fokker-Planck est une \u00E9quation aux d\u00E9riv\u00E9es partielles lin\u00E9aire que doit satisfaire la densit\u00E9 de probabilit\u00E9 de transition d'un processus de Markov. \u00C0 l'origine, une forme simplifi\u00E9e de cette \u00E9quation a permis d'\u00E9tudier le mouvement brownien. Comme la plupart des \u00E9quations aux d\u00E9riv\u00E9es partielles, elle ne donne des solutions explicites que dans des cas bien particuliers portant \u00E0 la fois sur la forme de l'\u00E9quation, sur la forme du domaine o\u00F9 elle est \u00E9tudi\u00E9e (conditions r\u00E9fl\u00E9chissante ou absorbante pour les particules browniennes et forme de l'espace dans lequel elles sont confin\u00E9es par exemple). Elle est nomm\u00E9e en l'honneur d'Adriaan Fokker et de Max Planck, les premiers physiciens \u00E0 l'avoir propos\u00E9e."@fr . . . . . "33220"^^ . "Equazione di Fokker-Planck"@it . "Fokker-Planck-Gleichung"@de . . . . . . "Fokker\u2013Planck equation"@en . . . . "\u30D5\u30A9\u30C3\u30AB\u30FC\u30FB\u30D7\u30E9\u30F3\u30AF\u65B9\u7A0B\u5F0F\uFF08\u82F1: Fokker\u2013Planck equation\uFF09\u3068\u306F\u3001\u7D71\u8A08\u529B\u5B66\u3067\u306B\u304A\u3044\u3066n \u2265 3 \u306E\u9805\u306E\u306A\u3044\u6B21\u306E\u65B9\u7A0B\u5F0F\u306E\u3053\u3068\u3092\u3044\u3046\u3002 \u7269\u7406\u91CFx (t) \u306E\u304C\u78BA\u7387\u5FAE\u5206\u65B9\u7A0B\u5F0F \u3068\u3044\u3046\u5F62\u3067\u4E0E\u3048\u3089\u308C\u308B\u3068\u3059\u308B\u3002\u305F\u3060\u3057\u3001R (t) \u306F\u767D\u8272\u96D1\u97F3\u306E\u30AC\u30A6\u30B9\u904E\u7A0B\uFF1A \u3067\u3042\u308B\u3002\u3053\u306E\u3068\u304D\u3001x \u306E\u78BA\u7387\u5206\u5E03P (x, t) \u306F\u30D5\u30A9\u30C3\u30AB\u30FC\u30FB\u30D7\u30E9\u30F3\u30AF\u65B9\u7A0B\u5F0F\u306B\u5F93\u3046\u3002\u305F\u3060\u3057\u4FC2\u6570\u306E\u5B9A\u7FA9\u306B\u306F\u4EE5\u4E0B\u306E2\u3064\u306E\u6D41\u5100\u304C\u3042\u308B\uFF1A \n* \u4F0A\u85E4\u6E05\u306E\u65B9\u6CD5 \n* \u306E\u65B9\u6CD5 \u7279\u306B\u7DDA\u5F62\u30D6\u30E9\u30A6\u30F3\u904B\u52D5\uFF08\u30AA\u30EB\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\uFF1D\u30A6\u30FC\u30EC\u30F3\u30D9\u30C3\u30AF\u904E\u7A0B\uFF09\u306B\u5BFE\u3059\u308B\u65B9\u7A0B\u5F0F\u3092\u7DDA\u5F62\u30D5\u30A9\u30C3\u30AB\u30FC\u30FB\u30D7\u30E9\u30F3\u30AF\u65B9\u7A0B\u5F0F\u3068\u3044\u3046\u3002\u3053\u306E\u3068\u304D\u306F \u3068\u306A\u308B\uFF08\u03B3 , D \u306F\u5B9A\u6570\uFF09\u3002\u3053\u308C\u306F \u3068\u3044\u3046\u30E9\u30F3\u30B8\u30E5\u30D0\u30F3\u65B9\u7A0B\u5F0F\u306B\u5BFE\u5FDC\u3059\u308B\u3002"@ja . . "In matematica e nella teoria della probabilit\u00E0, l'equazione di Fokker-Planck, il cui nome \u00E8 dovuto a Adriaan Fokker e a Max Planck, detta anche equazione anticipativa di Kolmogorov, descrive l'evoluzione temporale della funzione di densit\u00E0 di probabilit\u00E0 della posizione di una particella, e pu\u00F2 essere generalizzata ad altri enti osservabili. Il primo impiego dell'equazione di Fokker-Planck fu la descrizione statistica del moto browniano di una particella in un fluido. In una dimensione spaziale , l'equazione di Fokker-Planck per un processo con termine di deriva e termine di diffusione \u00E8:"@it . "\u0423\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0424\u043E\u043A\u043A\u0435\u0440\u0430 \u2014 \u041F\u043B\u0430\u043D\u043A\u0430 \u2014 \u043E\u0434\u043D\u043E \u0438\u0437 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0445 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0439 \u0432 \u0447\u0430\u0441\u0442\u043D\u044B\u0445 \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u044B\u0445, \u043E\u043F\u0438\u0441\u044B\u0432\u0430\u0435\u0442 \u0432\u0440\u0435\u043C\u0435\u043D\u043D\u0443\u0301\u044E \u044D\u0432\u043E\u043B\u044E\u0446\u0438\u044E \u0444\u0443\u043D\u043A\u0446\u0438\u0438 \u043F\u043B\u043E\u0442\u043D\u043E\u0441\u0442\u0438 \u0432\u0435\u0440\u043E\u044F\u0442\u043D\u043E\u0441\u0442\u0438 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0438 \u0438\u043C\u043F\u0443\u043B\u044C\u0441\u0430 \u0447\u0430\u0441\u0442\u0438\u0446 \u0432 \u043F\u0440\u043E\u0446\u0435\u0441\u0441\u0430\u0445, \u0433\u0434\u0435 \u0432\u0430\u0436\u043D\u0430 \u0441\u0442\u043E\u0445\u0430\u0441\u0442\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u043F\u0440\u0438\u0440\u043E\u0434\u0430 \u044F\u0432\u043B\u0435\u043D\u0438\u044F. \u041D\u0430\u0437\u0432\u0430\u043D\u043E \u0432 \u0447\u0435\u0441\u0442\u044C \u043D\u0438\u0434\u0435\u0440\u043B\u0430\u043D\u0434\u0441\u043A\u043E\u0433\u043E \u0438 \u043D\u0435\u043C\u0435\u0446\u043A\u043E\u0433\u043E \u0444\u0438\u0437\u0438\u043A\u043E\u0432 \u0410\u0434\u0440\u0438\u0430\u043D\u0430 \u0424\u043E\u043A\u043A\u0435\u0440\u0430 \u0438 \u041C\u0430\u043A\u0441\u0430 \u041F\u043B\u0430\u043D\u043A\u0430, \u0442\u0430\u043A\u0436\u0435 \u0438\u0437\u0432\u0435\u0441\u0442\u043D\u043E \u043A\u0430\u043A \u043F\u0440\u044F\u043C\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u041A\u043E\u043B\u043C\u043E\u0433\u043E\u0440\u043E\u0432\u0430. \u041C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u043E\u0431\u043E\u0431\u0449\u0435\u043D\u043E \u043D\u0430 \u0434\u0440\u0443\u0433\u0438\u0435 \u0438\u0437\u043C\u0435\u0440\u0438\u043C\u044B\u0435 \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u044B: \u0440\u0430\u0437\u043C\u0435\u0440 (\u0432 \u0442\u0435\u043E\u0440\u0438\u0438 \u043A\u043E\u0430\u043B\u0435\u0441\u0446\u0435\u043D\u0446\u0438\u0438), \u043C\u0430\u0441\u0441\u0430 \u0438 \u0442. \u0434."@ru . . . . "\u00C9quation de Fokker-Planck"@fr . "\u0420\u0456\u0432\u043D\u044F\u0301\u043D\u043D\u044F \u0424\u043E\u0301\u043A\u043A\u0435\u0440\u0430 \u2014 \u041F\u043B\u0430\u0301\u043D\u043A\u0430 \u2014 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0432 \u0447\u0430\u0441\u0442\u0438\u043D\u043D\u0438\u0445 \u043F\u043E\u0445\u0456\u0434\u043D\u0438\u0445, \u0449\u043E \u043E\u043F\u0438\u0441\u0443\u0454 \u0435\u0432\u043E\u043B\u044E\u0446\u0456\u044E \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0432\u0438\u043F\u0430\u0434\u043A\u043E\u0432\u043E\u0457 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0438. \u0414\u043B\u044F \u043E\u0434\u043D\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0432\u0438\u043F\u0430\u0434\u043A\u043E\u0432\u043E\u0457 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0438 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0424\u043E\u043A\u043A\u0435\u0440\u0430-\u041F\u043B\u0430\u043D\u043A\u0430 \u043C\u0430\u0454 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0438\u0439 \u0432\u0438\u0433\u043B\u044F\u0434 , \u0434\u0435 \u2014 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0432\u0438\u043F\u0430\u0434\u043A\u043E\u0432\u043E\u0457 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0438, \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0434\u0440\u0435\u0439\u0444\u043E\u0432\u0438\u043C \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u043E\u043C, \u0430 \u2014 \u0434\u0438\u0444\u0443\u0437\u0456\u0439\u043D\u0438\u043C \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u043E\u043C. \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u0443 \u0432\u0438\u043F\u0430\u0434\u043A\u0443 \u0431\u0440\u043E\u0443\u043D\u0456\u0432\u0441\u044C\u043A\u043E\u0433\u043E \u0440\u0443\u0445\u0443 \u0432\u0437\u0434\u043E\u0432\u0436 \u043F\u0440\u044F\u043C\u043E\u0457 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0424\u043E\u043A\u043A\u0435\u0440\u0430-\u041F\u043B\u0430\u043D\u043A\u0430 \u0434\u043B\u044F \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0447\u0430\u0441\u0442\u0438\u043D\u043E\u043A \u0437\u0430 \u0448\u0432\u0438\u0434\u043A\u043E\u0441\u0442\u044F\u043C\u0438 \u043C\u0430\u0454 \u0432\u0438\u0433\u043B\u044F\u0434: , \u0414\u0438\u0444\u0443\u0437\u0456\u0439\u043D\u0438\u0439 \u0456 \u0434\u0440\u0435\u0439\u0444\u043E\u0432\u0438\u0439 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0438 \u043C\u043E\u0436\u043D\u0430 \u043E\u0442\u0440\u0438\u043C\u0430\u0442\u0438, \u0440\u043E\u0437\u0433\u043B\u044F\u0434\u0430\u044E\u0447\u0438 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u041B\u0430\u043D\u0436\u0435\u0432\u0435\u043D\u0430."@uk . . . . . . . . . . . "\u30D5\u30A9\u30C3\u30AB\u30FC\u30FB\u30D7\u30E9\u30F3\u30AF\u65B9\u7A0B\u5F0F"@ja . . . . . . . . "Equa\u00E7\u00E3o de Fokker\u2013Planck"@pt . . . . . . "En mec\u00E1nica estad\u00EDstica, la ecuaci\u00F3n de Fokker\u2013Planck es una ecuaci\u00F3n diferencial parcial que describe la evoluci\u00F3n temporal de la funci\u00F3n de densidad de probabilidad de la velocidad de una part\u00EDcula bajo la influencia de fuerzas de arrastre y fuerzas aleatorias, como en el movimiento browniano. La ecuaci\u00F3n tambi\u00E9n puede generalizarse a otro tipo de variables.\u200B La ecuaci\u00F3n se aplica a sistemas que pueden ser descritos por un peque\u00F1o n\u00FAmero de \"macrovariables\", donde otros par\u00E1metros var\u00EDan tan r\u00E1pidamente con el tiempo que pueden ser tratados como \"ruido\" o una perturbaci\u00F3n."@es . . . "A equa\u00E7\u00E3o de Fokker\u2013Planck, denominada assim por Adriaan Fokker e Max Planck, e tamb\u00E9m conhecida como equa\u00E7\u00E3o avan\u00E7ada de Kolmog\u00F3rov (por Andr\u00E9i Kolmog\u00F3rov, quem primeiro a introduziu em um artigo de 1931 ), descreve a evolu\u00E7\u00E3o temporal da fun\u00E7\u00E3o de densidade de probabilidade que mostra a posi\u00E7\u00E3o e a velocidade de uma part\u00EDcula, ainda que possa ser generalizada a outro tipo de vari\u00E1veis. A equa\u00E7\u00E3o \u00E9 aplic\u00E1vel a sistemas que possam ser descritos por um pequeno n\u00FAmero de \"macrovari\u00E1veis\", onde outros par\u00E2metros variam t\u00E3o rapidamente com o tempo que podem ser tratados como \"ru\u00EDdo\" ou uma perturba\u00E7\u00E3o."@pt . "Ecuaci\u00F3n de Fokker-Planck"@es . . "\uD655\uB960 \uACFC\uC815 \uC774\uB860\uC5D0\uC11C, \uD3EC\uCEE4\uB974-\uD50C\uB791\uD06C \uBC29\uC815\uC2DD(Fokker-Planck\u65B9\u7A0B\u5F0F, \uC601\uC5B4: Fokker\u2013Planck equation)\uC740 \uC5B4\uB5A4 \uC774\uD1A0 \uD655\uB960 \uACFC\uC815\uC758 \uD655\uB960 \uBC00\uB3C4 \uD568\uC218\uAC00 \uB530\uB974\uB294 \uD3B8\uBBF8\uBD84 \uBC29\uC815\uC2DD\uC774\uB2E4. \uC774\uB294 \uC2DC\uAC04\uC5D0 \uB300\uD558\uC5EC 1\uCC28, \uACF5\uAC04\uC5D0 \uB300\uD558\uC5EC 2\uCC28 \uD3B8\uBBF8\uBD84 \uBC29\uC815\uC2DD\uC774\uB2E4. \uD615\uC2DD\uC801\uC73C\uB85C, \uC288\uB8B0\uB529\uAC70 \uBC29\uC815\uC2DD\uC758 \uC758 \uAF34\uC774\uB2E4."@ko . . "En mec\u00E1nica estad\u00EDstica, la ecuaci\u00F3n de Fokker\u2013Planck es una ecuaci\u00F3n diferencial parcial que describe la evoluci\u00F3n temporal de la funci\u00F3n de densidad de probabilidad de la velocidad de una part\u00EDcula bajo la influencia de fuerzas de arrastre y fuerzas aleatorias, como en el movimiento browniano. La ecuaci\u00F3n tambi\u00E9n puede generalizarse a otro tipo de variables.\u200B La ecuaci\u00F3n se aplica a sistemas que pueden ser descritos por un peque\u00F1o n\u00FAmero de \"macrovariables\", donde otros par\u00E1metros var\u00EDan tan r\u00E1pidamente con el tiempo que pueden ser tratados como \"ruido\" o una perturbaci\u00F3n. Fue nombrada en reconocimiento de Adriaan Fokker\u200B y Max Planck,\u200B y tambi\u00E9n es conocida como ecuaci\u00F3n avanzada de Kolmog\u00F3rov (difusi\u00F3n) (por Andr\u00E9i Kolmog\u00F3rov, que la introdujo por primera vez en un art\u00EDculo de 1931\u200B). Cuando se aplica a distribuciones de posici\u00F3n de part\u00EDculas, es m\u00E1s conocida como ecuaci\u00F3n de Smoluchowski. El caso de la difusi\u00F3n cero es conocido en mec\u00E1nica estad\u00EDstica como . La primera derivaci\u00F3n consistente de la ecuaci\u00F3n de Fokker-Planck en el esquema sencillo de la mec\u00E1nica cl\u00E1sica y cu\u00E1ntica fue realizado\u200B por los sovi\u00E9ticos y .\u200B"@es . . "R\u00F3wnanie Fokkera-Plancka \u2013 r\u00F3wnanie r\u00F3\u017Cniczkowe cz\u0105stkowe drugiego rz\u0119du. Opisuje ewolucj\u0119 czasow\u0105 funkcji g\u0119sto\u015Bci prawdopodobie\u0144stwa po\u0142o\u017Cenia i pr\u0119dko\u015Bci. Nazwa pochodzi od nazwisk i Maxa Plancka. Znane jest r\u00F3wnie\u017C pod nazw\u0105 prospektywnego . Po raz pierwszy r\u00F3wnanie to zosta\u0142o u\u017Cyte do opisu zjawiska ruch\u00F3w Browna cz\u0105stki zanurzonej w cieczy. Og\u00F3lna forma r\u00F3wnania dla N zmiennych: gdzie to wektor dryftu, a oznacza tensor dyfuzji."@pl . . "\u0423\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0424\u043E\u043A\u043A\u0435\u0440\u0430 \u2014 \u041F\u043B\u0430\u043D\u043A\u0430 \u2014 \u043E\u0434\u043D\u043E \u0438\u0437 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0445 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0439 \u0432 \u0447\u0430\u0441\u0442\u043D\u044B\u0445 \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u044B\u0445, \u043E\u043F\u0438\u0441\u044B\u0432\u0430\u0435\u0442 \u0432\u0440\u0435\u043C\u0435\u043D\u043D\u0443\u0301\u044E \u044D\u0432\u043E\u043B\u044E\u0446\u0438\u044E \u0444\u0443\u043D\u043A\u0446\u0438\u0438 \u043F\u043B\u043E\u0442\u043D\u043E\u0441\u0442\u0438 \u0432\u0435\u0440\u043E\u044F\u0442\u043D\u043E\u0441\u0442\u0438 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0438 \u0438\u043C\u043F\u0443\u043B\u044C\u0441\u0430 \u0447\u0430\u0441\u0442\u0438\u0446 \u0432 \u043F\u0440\u043E\u0446\u0435\u0441\u0441\u0430\u0445, \u0433\u0434\u0435 \u0432\u0430\u0436\u043D\u0430 \u0441\u0442\u043E\u0445\u0430\u0441\u0442\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u043F\u0440\u0438\u0440\u043E\u0434\u0430 \u044F\u0432\u043B\u0435\u043D\u0438\u044F. \u041D\u0430\u0437\u0432\u0430\u043D\u043E \u0432 \u0447\u0435\u0441\u0442\u044C \u043D\u0438\u0434\u0435\u0440\u043B\u0430\u043D\u0434\u0441\u043A\u043E\u0433\u043E \u0438 \u043D\u0435\u043C\u0435\u0446\u043A\u043E\u0433\u043E \u0444\u0438\u0437\u0438\u043A\u043E\u0432 \u0410\u0434\u0440\u0438\u0430\u043D\u0430 \u0424\u043E\u043A\u043A\u0435\u0440\u0430 \u0438 \u041C\u0430\u043A\u0441\u0430 \u041F\u043B\u0430\u043D\u043A\u0430, \u0442\u0430\u043A\u0436\u0435 \u0438\u0437\u0432\u0435\u0441\u0442\u043D\u043E \u043A\u0430\u043A \u043F\u0440\u044F\u043C\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u041A\u043E\u043B\u043C\u043E\u0433\u043E\u0440\u043E\u0432\u0430. \u041C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u043E\u0431\u043E\u0431\u0449\u0435\u043D\u043E \u043D\u0430 \u0434\u0440\u0443\u0433\u0438\u0435 \u0438\u0437\u043C\u0435\u0440\u0438\u043C\u044B\u0435 \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u044B: \u0440\u0430\u0437\u043C\u0435\u0440 (\u0432 \u0442\u0435\u043E\u0440\u0438\u0438 \u043A\u043E\u0430\u043B\u0435\u0441\u0446\u0435\u043D\u0446\u0438\u0438), \u043C\u0430\u0441\u0441\u0430 \u0438 \u0442. \u0434."@ru . "\uD655\uB960 \uACFC\uC815 \uC774\uB860\uC5D0\uC11C, \uD3EC\uCEE4\uB974-\uD50C\uB791\uD06C \uBC29\uC815\uC2DD(Fokker-Planck\u65B9\u7A0B\u5F0F, \uC601\uC5B4: Fokker\u2013Planck equation)\uC740 \uC5B4\uB5A4 \uC774\uD1A0 \uD655\uB960 \uACFC\uC815\uC758 \uD655\uB960 \uBC00\uB3C4 \uD568\uC218\uAC00 \uB530\uB974\uB294 \uD3B8\uBBF8\uBD84 \uBC29\uC815\uC2DD\uC774\uB2E4. \uC774\uB294 \uC2DC\uAC04\uC5D0 \uB300\uD558\uC5EC 1\uCC28, \uACF5\uAC04\uC5D0 \uB300\uD558\uC5EC 2\uCC28 \uD3B8\uBBF8\uBD84 \uBC29\uC815\uC2DD\uC774\uB2E4. \uD615\uC2DD\uC801\uC73C\uB85C, \uC288\uB8B0\uB529\uAC70 \uBC29\uC815\uC2DD\uC758 \uC758 \uAF34\uC774\uB2E4."@ko . "\uD3EC\uCEE4\uB974-\uD50C\uB791\uD06C \uBC29\uC815\uC2DD"@ko . . . . "\u798F\u514B-\u666E\u6717\u514B\u65B9\u7A0B"@zh . . . . . . "\u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631 \u0628\u0644\u0627\u0646\u0643 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Fokker-Planck equation)\u200F \u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0635\u0641 \u0627\u0644\u062A\u0637\u0648\u0631 \u0627\u0644\u0632\u0645\u0646\u064A \u00AB\u0627\u0644\u062A\u063A\u064A\u0631 \u062E\u0644\u0627\u0644 \u0627\u0644\u0632\u0645\u0646\u00BB \u0644\u062A\u0627\u0628\u0639 \u0627\u0644\u0643\u062B\u0627\u0641\u0629 \u0627\u0644\u0627\u062D\u062A\u0645\u0627\u0644\u064A\u0629 \u0644\u0633\u0631\u0639\u0629 \u062C\u0633\u064A\u0645 \u0645\u0627. \u0648\u064A\u0645\u0643\u0646 \u0627\u0646 \u062A\u0643\u062A\u0628 \u0628\u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u0639\u0627\u0645 \u0639\u0644\u0649 \u0634\u0643\u0644 \u0627\u0628\u0639\u0627\u062F \u0645\u062A\u0639\u062F\u062F\u0629 \u0628\u0627\u0633\u062A\u062E\u062F\u0627\u0645 \u0627\u0644\u0645\u0624\u062B\u0631 \u0646\u0628\u0644\u0627. \u0644\u0643\u0646\u0647\u0627 \u063A\u0627\u0644\u0628\u0627 \u0645\u0627 \u062A\u0643\u062A\u0628 \u0628\u0627\u062E\u062A\u0635\u0627\u0631 \u0639\u0644\u0649 \u0627\u0639\u062A\u0628\u0627\u0631 \u0627\u0644\u062D\u0631\u0643\u0629 \u0630\u0627\u062A \u0628\u0639\u062F \u0648\u0627\u062D\u062F. \u0648\u0642\u062F \u0623\u062C\u0631\u064A \u0623\u0648\u0644 \u0627\u0634\u062A\u0642\u0627\u0642 \u0645\u062C\u0647\u0631\u064A \u0645\u062A\u0633\u0642 \u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631 \u0628\u0644\u0627\u0646\u0643 \u0641\u064A \u0645\u062E\u0637\u0637 \u0648\u0627\u062D\u062F \u0645\u0646 \u0627\u0644\u0645\u064A\u0643\u0627\u0646\u064A\u0643\u0627 \u0627\u0644\u0643\u0644\u0627\u0633\u064A\u0643\u064A\u0629 \u0648\u0627\u0644\u0643\u0645\u064A\u0629 \u0628\u0648\u0627\u0633\u0637\u0629 \u0645\u0646 \u0642\u0628\u0644 \u0646\u064A\u0643\u0648\u0644\u0627\u064A \u0628\u0648\u063A\u0648\u0644\u064A\u0648\u0628\u0648\u0641 . \u0641\u064A \u0628\u0639\u062F \u0645\u0643\u0627\u0646\u064A \u0648\u0627\u062D\u062F x, \u0645\u0639\u0627\u062F\u0644\u0629 \u0641\u0648\u0643\u0631 \u2013 \u0628\u0644\u0627\u0646\u0643 \u0644\u0645\u0639\u0644\u0627\u062C \u0645\u0639 \u0627\u0646\u062C\u0631\u0627\u0641 \u0627\u064A\u062A\u0648 D1(x,t) \u0648\u0646\u0634\u0631 D2(x,t) \u0647\u0648: \u062D\u064A\u062B \u0647\u0648 \u0645\u062A\u062C\u0647 \u0627\u0644\u0627\u0646\u062C\u0631\u0627\u0641 \u0648 \u0647\u0648 \u0645\u0648\u062A\u0631 \u0627\u0644\u0627\u0646\u062A\u0634\u0627\u0631; \u0648\u064A\u0646\u062A\u062C \u0647\u0630\u0647 \u0627\u0644\u0623\u062E\u064A\u0631 \u0645\u0646 \u0648\u062C\u0648\u062F \u0627\u0644\u0642\u0648\u0629 \u0627\u0644\u062A\u0635\u0627\u062F\u0641\u064A\u0629.."@ar . . . . . . . . . . . "\u0420\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0424\u043E\u043A\u043A\u0435\u0440\u0430 \u2014 \u041F\u043B\u0430\u043D\u043A\u0430"@uk . . . . . "In statistical mechanics, the Fokker\u2013Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection\u2013diffusion equation. The case with zero diffusion is the continuity equation. The Fokker\u2013Planck equation is obtained from the master equation through Kramers\u2013Moyal expansion. The first consistent microscopic derivation of the Fokker\u2013Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov."@en . "1118571382"^^ . . . . . "R\u00F3wnanie Fokkera-Plancka \u2013 r\u00F3wnanie r\u00F3\u017Cniczkowe cz\u0105stkowe drugiego rz\u0119du. Opisuje ewolucj\u0119 czasow\u0105 funkcji g\u0119sto\u015Bci prawdopodobie\u0144stwa po\u0142o\u017Cenia i pr\u0119dko\u015Bci. Nazwa pochodzi od nazwisk i Maxa Plancka. Znane jest r\u00F3wnie\u017C pod nazw\u0105 prospektywnego . Po raz pierwszy r\u00F3wnanie to zosta\u0142o u\u017Cyte do opisu zjawiska ruch\u00F3w Browna cz\u0105stki zanurzonej w cieczy. Og\u00F3lna forma r\u00F3wnania dla N zmiennych: gdzie to wektor dryftu, a oznacza tensor dyfuzji."@pl . . . . . . . . . . . "\u30D5\u30A9\u30C3\u30AB\u30FC\u30FB\u30D7\u30E9\u30F3\u30AF\u65B9\u7A0B\u5F0F\uFF08\u82F1: Fokker\u2013Planck equation\uFF09\u3068\u306F\u3001\u7D71\u8A08\u529B\u5B66\u3067\u306B\u304A\u3044\u3066n \u2265 3 \u306E\u9805\u306E\u306A\u3044\u6B21\u306E\u65B9\u7A0B\u5F0F\u306E\u3053\u3068\u3092\u3044\u3046\u3002 \u7269\u7406\u91CFx (t) \u306E\u304C\u78BA\u7387\u5FAE\u5206\u65B9\u7A0B\u5F0F \u3068\u3044\u3046\u5F62\u3067\u4E0E\u3048\u3089\u308C\u308B\u3068\u3059\u308B\u3002\u305F\u3060\u3057\u3001R (t) \u306F\u767D\u8272\u96D1\u97F3\u306E\u30AC\u30A6\u30B9\u904E\u7A0B\uFF1A \u3067\u3042\u308B\u3002\u3053\u306E\u3068\u304D\u3001x \u306E\u78BA\u7387\u5206\u5E03P (x, t) \u306F\u30D5\u30A9\u30C3\u30AB\u30FC\u30FB\u30D7\u30E9\u30F3\u30AF\u65B9\u7A0B\u5F0F\u306B\u5F93\u3046\u3002\u305F\u3060\u3057\u4FC2\u6570\u306E\u5B9A\u7FA9\u306B\u306F\u4EE5\u4E0B\u306E2\u3064\u306E\u6D41\u5100\u304C\u3042\u308B\uFF1A \n* \u4F0A\u85E4\u6E05\u306E\u65B9\u6CD5 \n* \u306E\u65B9\u6CD5 \u7279\u306B\u7DDA\u5F62\u30D6\u30E9\u30A6\u30F3\u904B\u52D5\uFF08\u30AA\u30EB\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\uFF1D\u30A6\u30FC\u30EC\u30F3\u30D9\u30C3\u30AF\u904E\u7A0B\uFF09\u306B\u5BFE\u3059\u308B\u65B9\u7A0B\u5F0F\u3092\u7DDA\u5F62\u30D5\u30A9\u30C3\u30AB\u30FC\u30FB\u30D7\u30E9\u30F3\u30AF\u65B9\u7A0B\u5F0F\u3068\u3044\u3046\u3002\u3053\u306E\u3068\u304D\u306F \u3068\u306A\u308B\uFF08\u03B3 , D \u306F\u5B9A\u6570\uFF09\u3002\u3053\u308C\u306F \u3068\u3044\u3046\u30E9\u30F3\u30B8\u30E5\u30D0\u30F3\u65B9\u7A0B\u5F0F\u306B\u5BFE\u5FDC\u3059\u308B\u3002"@ja . . . . "\u798F\u514B-\u666E\u6717\u514B\u65B9\u7A0B\uFF08Fokker\u2013Planck equation\uFF09\u63CF\u8FF0\u7C92\u5B50\u5728\u4F4D\u80FD\u5834\u4E2D\u53D7\u5230\u96A8\u6A5F\u529B\u5F8C\uFF0C\u96A8\u6642\u9593\u6F14\u5316\u7684\u4F4D\u7F6E\u6216\u662F\u901F\u5EA6\u7684\u5206\u5E03\u51FD\u6578 \u3002\u6B64\u65B9\u7A0B\u5F0F\u4EE5\u8377\u862D\u7269\u7406\u5B78\u5BB6\u963F\u5FB7\u91CC\u5B89\u00B7\u798F\u514B\u8207\u99AC\u514B\u65AF\u00B7\u666E\u6717\u514B\u7684\u59D3\u6C0F\u4F86\u547D\u540D\u3002 \u4E00\u7DAD x\u65B9\u5411\u4E0A,\u798F\u514B-\u666E\u6717\u514B\u65B9\u7A0B\u6709\u5169\u500B\u53C3\u6578\uFF0C\u4E00\u662F\u62D6\u66F3\u53C3\u6578 D1(x,t)\uFF0C\u53E6\u4E00\u662F\u64F4\u6563 D2(x,t) \u5728 \u7DAD\u7A7A\u9593\u4E2D\u7684\u798F\u514B-\u666E\u6717\u514B\u65B9\u7A0B\u662F \u662F\u7B2C\u7DAD\u5EA6\u7684\u4F4D\u7F6E\uFF0C\u6B64\u6642 \u70BA\u62D6\u66F3\u5411\u91CF\uFF0C\u70BA\u64F4\u6563\u5F35\u91CF\u3002"@zh . . . "Die Fokker-Planck-Gleichung (FPG, nach Adriaan Dani\u00EBl Fokker (1887\u20131972) und Max Planck (1858\u20131947)) ist eine partielle Differentialgleichung. Sie beschreibt die zeitliche Entwicklung einer Wahrscheinlichkeitsdichtefunktion unter der Wirkung von Drift und Diffusion . In ihrer eindimensionalen Form lautet die Gleichung: In der Wahrscheinlichkeitstheorie ist diese Gleichung auch bekannt als Kolmogorov-Vorw\u00E4rtsgleichung und in diesem Fall nach dem Mathematiker Andrei Nikolajewitsch Kolmogorow benannt. Sie ist eine lineare parabolische partielle Differentialgleichung, die sich nur f\u00FCr einige Spezialf\u00E4lle (einfache K\u00F6rpergeometrie; Linearit\u00E4t der Randbedingungen, des Drift- und des Diffusionskoeffizienten) analytisch exakt l\u00F6sen l\u00E4sst. F\u00FCr verschwindende Drift und konstante Diffusion geht die FPG in die Diffusions- (oder auch W\u00E4rmeleitungs-) Gleichung \u00FCber. In Dimensionen lautet die Fokker-Planck-Gleichung Von der Smoluchowski-Gleichung spricht man, wenn die Positionen der Teilchen im System beschreibt. F\u00FCr Markovsche Prozesse geht die FPG aus der Kramers-Moyal-Entwicklung hervor, die nach der zweiten Ordnung abgebrochen wird. Von gro\u00DFer Bedeutung ist die \u00E4quivalente Beschreibung von Problemen durch Langevin-Gleichungen, die im Vergleich zur FPG die mikroskopische Dynamik stochastischer Systeme beschreiben und \u2013 im Gegensatz zur FPG \u2013 im Allgemeinen nichtlinear sind."@de . . . "In statistical mechanics, the Fokker\u2013Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The first consistent microscopic derivation of the Fokker\u2013Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov."@en . . . . . . . "166896"^^ . "L'\u00E9quation de Fokker-Planck est une \u00E9quation aux d\u00E9riv\u00E9es partielles lin\u00E9aire que doit satisfaire la densit\u00E9 de probabilit\u00E9 de transition d'un processus de Markov. \u00C0 l'origine, une forme simplifi\u00E9e de cette \u00E9quation a permis d'\u00E9tudier le mouvement brownien. Comme la plupart des \u00E9quations aux d\u00E9riv\u00E9es partielles, elle ne donne des solutions explicites que dans des cas bien particuliers portant \u00E0 la fois sur la forme de l'\u00E9quation, sur la forme du domaine o\u00F9 elle est \u00E9tudi\u00E9e (conditions r\u00E9fl\u00E9chissante ou absorbante pour les particules browniennes et forme de l'espace dans lequel elles sont confin\u00E9es par exemple). Elle est nomm\u00E9e en l'honneur d'Adriaan Fokker et de Max Planck, les premiers physiciens \u00E0 l'avoir propos\u00E9e."@fr .