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Separation of variables in the Kramers equation

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Renat Zhdanov, Alexander Zhalij

Mathematical Physics · Submitted 4 Jun 1999

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is obtained.

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arXiv IDmath-ph/9906004 (version 1)
Pith IDD5XLSLDO, Pith's permanent record of this paper and its review
CitationAlexander Zhalij, Renat Zhdanov. "Separation of variables in the Kramers equation." arXiv:math-ph/9906004 (1999). https://pith.science/paper/D5XLSLDO
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,855 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.83 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group. Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is obtained. Furthermore, we construct solutions of the Kramers equation with separated variables in explicit form.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group. background 0.74 0.55 0.09 asserted 0.63
s1 Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is obtained. result 0.82 0.89 0.58 asserted 1.00
s2 Furthermore, we construct solutions of the Kramers equation with separated variables in explicit form. contribution 0.64 0.95 0.27 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group. Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is obtained. Furthermore, we construct solutions of the Kramers equation with separated variables in explicit form.

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