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  Название: A Soluble Model for Diffusion in a Bistable Potential
  Автор: van Kampen N.G.
  After a general discussion of diffusion in a potential field having two minima, an explicit solution is obtained for a special form of the potential. The potential is symmetric, but the diffusing particle is initially at an arbitrary point near the potential maximum. Various suggested approximation schemes are tested, with the following conclusions, (i) In linear approximation around the maximum the probability distribution is Gaussian. A two-peaked distribution emerges only through nonlinear terms, (ii) The chances for the particle to go to the right or the left valley cannot be found from the linear approximation. Matching the linear approximation with a macroscopic description far away from the maximum is therefore wrong, (iii) Kramers' treatment of the escape across a potential barrier yields a practically exact result for this model.
explanation: The text is an abstract or summary of a scientific paper titled "A Soluble Model for Diffusion in a Bistable Potential" by N.G. van Kampen. It discusses the diffusion of a particle in a potential field with two minima, focusing on the derivation of an explicit solution for a specific potential form. The text does not contain a math question or an answer to a math question.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Нашли опечатку? Выделите ее мышкой и нажмите Ctrl+Enter
  Название: A Soluble Model for Diffusion in a Bistable Potential
  Автор: van Kampen N.G.
  After a general discussion of diffusion in a potential field having two minima, an explicit solution is obtained for a special form of the potential. The potential is symmetric, but the diffusing particle is initially at an arbitrary point near the potential maximum. Various suggested approximation schemes are tested, with the following conclusions, (i) In linear approximation around the maximum the probability distribution is Gaussian. A two-peaked distribution emerges only through nonlinear terms, (ii) The chances for the particle to go to the right or the left valley cannot be found from the linear approximation. Matching the linear approximation with a macroscopic description far away from the maximum is therefore wrong, (iii) Kramers' treatment of the escape across a potential barrier yields a practically exact result for this model.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text is an abstract or summary of a scientific paper titled "A Soluble Model for Diffusion in a Bistable Potential" by N.G. van Kampen. It discusses the diffusion of a particle in a potential field with two minima, focusing on the derivation of an explicit solution for a specific potential form. The text does not contain a math question or an answer to a math question.
  
  Label: no-question