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Updated: Sep 22, 2025

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Random-matrix approach to transition-state theory.

H A Weidenmüller1

  • 1Max-Planck-Institut für Kernphysik, D-69029 Heidelberg, Germany.

Physical Review. E
|May 20, 2022
PubMed
Summary

This study models complex systems separated by barriers using random Hamiltonians. The findings confirm general transition-state theory, applicable even for thick barriers with relaxed conditions.

Area of Science:

  • Quantum mechanics
  • Statistical physics

Background:

  • Complex systems often feature barriers, hindering transitions.
  • Modeling these systems requires understanding inter-channel dynamics.

Purpose of the Study:

  • To develop a model for systems separated by barriers.
  • To calculate average transition probabilities between coupled Hamiltonians.
  • To validate transition-state theory.

Main Methods:

  • Utilizing two random Hamiltonians coupled via tunneling or a transition state.
  • Analyzing the model in the universal limit of large matrix dimension.
  • Calculating average transition probabilities (⟨P_{ab}⟩).

Main Results:

  • Derived the formula ⟨P_{ab}⟩ = P_{a}T_{b}/∑_{b^{'}}T_{b^{'}} under the condition ∑_{b^{'}}T_{b^{'}}≫1.

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  • P_{a} represents the probability of forming the tunneling channel or transition state.
  • T_{b^{'}} are transmission coefficients for channels coupled to the second Hamiltonian.

    • Conclusions:

    • The derived formula confirms transition-state theory in its general form.
    • The theory's applicability extends to thick barriers where the condition is relaxed.
    • This provides a more general framework for understanding tunneling processes.