Quantum and Classical Dynamics Extend Stochastic Modelling for Out-of-Equilibrium Systems
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Understanding the behaviour of systems far from equilibrium presents a significant challenge in modern nanoscale technologies, with implications for fields ranging from molecular electronics to computation. Stefano Giordano, Giuseppe Florio, and colleagues, including Giuseppe Puglisi, Fabrizio Cleri, and Ralf Blossey, now present a unified framework bridging classical and quantum descriptions of these open systems. Their work addresses the need for dynamical models that respect both classical laws and the principles of quantum mechanics, specifically complete positivity. By extending classical stochastic dynamics and formulating a generalized Langevin equation, the researchers derive master equations that reveal a surprising universality, demonstrating that complete positivity requires symmetrical treatment of friction and noise within both classical and quantum formulations, and offering a powerful tool for analysing a broad range of non-equilibrium nanoscale systems. Quantum Foundations, Thermodynamics and Information Theory This extensive compilation covers a broad spectrum of topics within statistical mechanics, quantum mechanics, stochastic processes, and related mathematical tools. The collection focuses on foundational texts, entropy and the second law, non-equilibrium thermodynamics, fluctuation theorems, and stochastic energetics, providing a comprehensive resource for researchers in these fields. Key works include Gibbs’ foundational principles of statistical mechanics and Landau and Lifshitz’s contributions, which integrate statistical mechanics with quantum mechanics. Klein’s historical paper on the quantum mechanical foundation of the second law and Vedral’s work on relative entropy, crucial for quantum information theory, are also prominently featured. Researchers extensively explore non-equilibrium thermodynamics, with Seifert’s work serving as a highly influential contribution to stochastic thermodynamics and fluctuation theorems. Tomé and de Oliveira have also made significant contributions to this area, furthering our understanding of systems driven away from equilibrium. The collection also delves into stochastic processes and Langevin equations, providing comprehensive treatments from Van Kampen, Risken, and Coffey, Kalmykov, and Waldron. Researchers examine the Langevin equation itself, investigating its derivation and applications through the contributions of Netz and Muradore, while also exploring differing interpretations of stochastic integrals, as discussed by Sokolov. The compilation extends to the realm of quantum mechanics and open quantum systems, featuring standard textbooks by Cohen-Tannoudji, Diu, and Laloe, and Breuer and Petruccione. Englert and Morigi, Gardiner and Zoller, and Breuer and Petruccione provide key references for understanding how quantum systems interact with their environment. Researchers explore the quantum version of the Fokker-Planck equation, connecting it to the Lindblad equation through the work of de Oliveira, and investigate the quantum mechanical treatment of friction and noise, as presented by Dekker and Scutaru. The collection also includes essential mathematical tools, such as matrix analysis from Gantmacher and Lancaster and Tismenetsky, and covers various types of differential equations. Specific areas and emerging themes are also addressed, including the stochastic thermodynamics of holonomic systems, explored by Giordano, and effective diffusion constants, investigated by Giordano and Blossey. This compilation demonstrates the strong connections between statistical mechanics, quantum mechanics, stochastic processes, and mathematics, with a particular focus on understanding systems out of equilibrium. Langevin and Klein-Kramers Equations for Open Systems Scientists have developed a new framework for modeling open quantum systems, addressing a critical challenge in nanoscale technologies such as molecular electronics and computation. The work begins with a generalized Langevin equation, where frictional forces and noise act symmetrically on the system’s equations of motion, a departure from traditional approaches. From this equation, researchers derived a generalized Klein-Kramers equation, expressed using Poisson brackets, and demonstrated that its solutions converge to the classical canonical distribution, ensuring consistency with established thermodynamic principles. To rigorously verify this consistency, the team defined heat and entropy along individual trajectories, confirming adherence to both the first and second laws of thermodynamics. Building upon this classical foundation, scientists transitioned to the quantum realm by applying canonical quantization, involving the replacement of Poisson brackets with commutators, to the Klein-Kramers equation. This process introduced friction operators, defined in two distinct ways, Hermitian and non-Hermitian, leading to two separate master equations describing the system’s quantum dynamics. Analysis of a harmonic oscillator revealed that these master equations reduce to a Lindblad-type generator, guaranteeing complete positivity, only when friction and noise are included symmetrically in both equations of motion, fully justifying the initial classical construction.
Results demonstrate that only models incorporating symmetric friction and noise consistently preserve the positivity of the density matrix and adhere to thermodynamic principles, confirming the validity and robustness of the developed formalism. Thermodynamic Consistency in Open Quantum Systems Scientists have developed a novel framework for understanding open quantum systems, extending classical stochastic dynamics to the quantum domain. The work begins with a generalized Langevin equation, incorporating both friction and noise symmetrically within the equations of motion, a crucial step in ensuring thermodynamic consistency. From this, researchers derived a generalized Klein-Kramers equation, expressed using Poisson brackets, and demonstrated that its asymptotic solution aligns with the classical canonical distribution, confirming its adherence to established statistical mechanics.
The team demonstrated full consistency with both the first and second laws of thermodynamics along individual trajectories, validating the approach’s physical realism. Applying canonical quantization, they obtained two distinct quantum master equations, differentiated by whether friction operators were treated as Hermitian or non-Hermitian. Analysis of a harmonic oscillator revealed that these equations reduce to a Lindblad-type generator, a key indicator of complete positivity, only when friction and noise are included in both equations of motion. This finding fully justifies the initial symmetrical inclusion of friction and noise in the classical construction. Further investigation showed that the friction coefficients must satisfy the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a universality that transcends the specific operator representation used. This formalism provides a versatile tool for deriving quantum versions of thermodynamic laws and is directly applicable to a wide range of nonequilibrium nanoscale systems. Unified Classical and Quantum Open System Dynamics This work presents a unified framework bridging classical and quantum descriptions of open systems, building upon a generalized Langevin dynamics where both friction and noise symmetrically influence the equations governing position and momentum. The resulting classical Klein-Kramers equation preserves the established Gibbs distribution and adheres to the fundamental laws of thermodynamics, demonstrating the importance of incorporating both dissipative and stochastic elements for physical consistency. Applying canonical quantization to this framework yields two distinct quantum master equations, each linked to different representations of friction operators, yet both consistently reduce to Lindblad-type dynamics for a harmonic oscillator under a shared constraint on friction coefficients. The research demonstrates that complete positivity in the quantum realm aligns with the classical requirement of symmetric dissipation. 👉 More information 🗞 Boltzmann to Lindblad: Classical and Quantum Approaches to Out-of-Equilibrium Statistical Mechanics 🧠 ArXiv: https://arxiv.org/abs/2512.11613 Tags: