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This summary is machine-generated.

This study presents a new closed-form solution for analyzing relaxation dynamics in statistical physics models. The method simplifies solving complex recurrence relations, aiding in understanding system behavior and identifying eigenspectra.

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Area of Science:

  • Statistical Physics
  • Economics
  • Mathematical Physics

Background:

  • Statistical physics explores system relaxation to equilibrium, crucial in economics with extrinsic noise.
  • Agent-based models often simplify complex dynamics, necessitating analytical solutions for relaxation processes.

Purpose of the Study:

  • To derive a concise closed-form solution for a linear three-term recurrence relation.
  • To provide an efficient method for analyzing relaxation dynamics in binary choice models and other stochastic processes.

Main Methods:

  • Employed a linear algebraic approach using properties of lower-triangular and tridiagonal matrices.
  • Utilized orthogonal polynomials to express terms in the recurrence relation.
  • Focused on power series coefficients of Heun functions.

Main Results:

  • Derived a closed-form solution to the three-term recurrence relation, bypassing traditional continued fractions.
  • Applied the solution to social choice models, yielding equations for relaxation to steady-state.
  • Enabled easy identification of eigenspectra for one-dimensional, one-step, continuous-time Markov processes.

Conclusions:

  • The derived solution offers a novel, off-the-shelf method for analyzing complex stochastic processes.
  • This approach simplifies the study of relaxation dynamics in various scientific fields, including economics and quantum mechanics.