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If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.
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Normalized Gaussian path integrals.

Giulio Corazza1, Matteo Fadel2

  • 1Laboratory for Computation and Visualization in Mathematics and Mechanics (LCVMM) Institute of Mathematics, Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland.

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This study presents a novel method for solving Gaussian path integrals, enabling the calculation of transition probabilities for systems with noise. The approach simplifies complex dynamics, offering insights into polymer physics and stochastic processes.

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Area of Science:

  • Mathematical Physics
  • Statistical Mechanics
  • Computational Physics

Background:

  • Path integrals are fundamental for modeling physical systems influenced by classical or quantum noise.
  • Normalized path integrals quantify the probability of transitioning between system states.
  • Existing methods for solving path integrals, especially in the presence of noise, can be computationally intensive.

Purpose of the Study:

  • To develop a consistent approach for solving Euclidean (Wiener) Gaussian path integrals.
  • To enable the computation of transition probabilities using the semiclassical approximation.
  • To provide a method applicable to Fokker-Planck dynamics and the physics of stringlike objects.

Main Methods:

  • Solving conditional and unconditional Euclidean Gaussian path integrals.
  • Utilizing the semiclassical approximation to compute transition probabilities.
  • Deriving solutions from a system of linear differential equations.

Main Results:

  • A consistent method for solving Gaussian path integrals is demonstrated.
  • Transition probabilities can be computed efficiently via linear differential equations.
  • The method is applied to derive the dynamics of the Ornstein-Uhlenbeck process and the Van der Pol oscillator.
  • The end-to-end transition probability for a charged string under an external field is calculated.

Conclusions:

  • The proposed method offers a computationally tractable way to determine transition probabilities in noisy systems.
  • This approach is particularly valuable for studying complex systems like polymers and stochastic processes.
  • The findings advance the understanding of dynamics in systems governed by Fokker-Planck equations.