[Paper Review] Inclusive Spectra and Quantum Stochastic Processes

A. Krzywicki|ArXiv.org|Apr 10, 2002

High-Energy Particle Collisions Research3 citations

TL;DR

This paper proposes a quantum stochastic model to explain the emergence of thermal-like inclusive spectra in high-energy hadronic collisions, even without genuine thermal equilibrium. By modeling a test particle's momentum evolution via a non-Hermitian master equation with complex eigenvalues, the system evolves toward a unique stationary state with a Boltzmann-like momentum distribution, suggesting that pseudo-thermalization of multiparton wave functions could underlie observed thermal spectra in hadron collisions.

ABSTRACT

We try to explain the apparently "thermal" shape of the inclusive transverse momentum spectrum. We conjecture that prior to the collision, parton-parton interactions generate a kind of a stochastic process driving the one-particle spectrum within a hadron towards the thermal shape it would have in a classical gas. We illustrate this idea, which might have a more general relevance, with a simple, exactly solvable model.

Motivation & Objective

• To investigate whether the one-particle spectrum of a many-body quantum system evolves toward a universal, thermal-like distribution independent of initial conditions.
• To explore if non-unitary, non-Hermitian evolution in a quantum system can mimic classical thermalization through stochastic-like dynamics.
• To provide a minimal, analytically solvable model that reproduces the key features of thermalization in a quantum setting.
• To connect this mechanism to the phenomenological observation of Boltzmann-like spectra in high-energy hadronic processes, such as inclusive transverse momentum distributions.
• To argue that pseudo-thermalization at the parton level could explain thermal-like spectra in hadron collisions where true thermalization is implausible.

Proposed method

• Formulates a classical stochastic process with transition probabilities satisfying detailed balance, leading to a Boltzmann equilibrium distribution.
• Introduces a non-Hermitian quantum master equation for the wave function amplitude, derived from a non-unitary interaction Hamiltonian.
• Assumes linear dependence of phase factors on energy differences to enable analytical solution of the eigenvalue problem.
• Transforms the discrete eigenvalue equation into momentum space via Fourier transform, yielding a cosine-type dispersion relation.
• Solves for eigenstates with complex exponents, resulting in wave functions that decay exponentially in energy and oscillate spatially.
• Demonstrates that the stationary state has a momentum distribution matching the Gibbs-Boltzmann form, independent of initial conditions.

Experimental results

Research questions

• RQ1 Can a pure quantum state of a many-body system evolve toward a universal one-particle momentum distribution resembling a thermal (Boltzmann) form?
• RQ2 Does non-unitary, non-Hermitian evolution in a quantum system lead to a unique stationary state independent of initial conditions, akin to a Markov process?
• RQ3 Can such a quantum stochastic process explain the observed thermal-like inclusive spectra in high-energy hadronic collisions where genuine thermalization is unlikely?
• RQ4 Is the emergence of a thermal-like spectrum in the one-particle sector a generic feature of multiparton wave functions under non-unitary evolution?
• RQ5 How does the structure of the interaction Hamiltonian (specifically its non-Hermitian nature) influence the convergence to a stationary state with thermal characteristics?

Key findings

• The quantum model exhibits a unique stationary state with a momentum distribution that matches the Gibbs-Boltzmann form, $ w(p) o e^{-E(p)/T} $, independent of initial conditions.
• The stationary state arises from complex eigenvalues of a non-Hermitian evolution generator, indicating a decay-like process toward a single quantum state.
• The wave function of the stationary state is $ ilde{ ho}(p) o e^{-E(p)(1+2i\xi_1)/2T} $, with a complex phase that breaks time-reversal symmetry.
• The model’s eigenvalue spectrum is $ \lambda_x = 2e^{i\xi_0} \cos(x\cdot\delta) $, showing that the system supports a continuous family of stationary modes.
• The stationary solution is robust under broad parameter choices, indicating that pseudo-thermalization is a generic feature of the model for a wide range of parameters.
• The result supports the idea that thermal-like inclusive spectra in high-energy hadron collisions may originate from pseudo-thermalization of multiparton wave functions, not from final-state thermalization.