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Han P, He G, Huang Z, Guo F. Semiconductor laser systems excited by multiplicative Ornstein-Uhlenbeck noise and additive sine-Wiener noise in relation to real and imaginary parts. Phys Rev E 2024; 109:064126. [PMID: 39020954 DOI: 10.1103/physreve.109.064126] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2023] [Accepted: 05/08/2024] [Indexed: 07/20/2024]
Abstract
In this paper we study a semiconductor laser system excited by multiplicative Ornstein-Uhlenbeck (OU) noise and additive sine-Wiener (SW) noise with a related real part and imaginary part. The two-dimensional Langevin equation (LE) with cross-correlating complex OU noise and complex SW noise is equivalent to the six-dimensional LE. Based on functional methods and spatial dimension extension methods, the six-dimensional Fokker-Planck equation (FPE) is achieved. Through Taylor expansion approximation and linear transformation, the dimensionality of FPE is reduced, and the exact expression of the probability distribution (SPD) in steady state is obtained. The impacts of noise parameters on laser intensity on SPD are further analyzed. Moreover, based on the reduced FPE, the information entropy of the system is obtained. This calculation is useful for us to understand further the influence of system parameters and noise correlation coefficient on the system. The main work lies in obtaining an alternative method for deriving a FPE corresponding to the two-dimensional stochastic semiconductor laser model, which is more applicable than methods in previous literature. An equivalent expression for cross-correlated OU noise and cross-correlated SW noise as well as a method for approximate solutions of a high-dimensional FPE are also provided.
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Martin D, O'Byrne J, Cates ME, Fodor É, Nardini C, Tailleur J, van Wijland F. Statistical mechanics of active Ornstein-Uhlenbeck particles. Phys Rev E 2021; 103:032607. [PMID: 33862678 DOI: 10.1103/physreve.103.032607] [Citation(s) in RCA: 68] [Impact Index Per Article: 22.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2020] [Accepted: 12/23/2020] [Indexed: 06/12/2023]
Abstract
We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.
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Affiliation(s)
- David Martin
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
| | - Jérémy O'Byrne
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
| | - Michael E Cates
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Étienne Fodor
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg
| | - Cesare Nardini
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- Service de Physique de l'État Condensé, CNRS UMR 3680, CEA-Saclay, 91191 Gif-sur-Yvette, France
| | - Julien Tailleur
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
| | - Frédéric van Wijland
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
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Abstract
We derive an analytic expression for the distribution of velocities of multiple interacting active particles which we test by numerical simulations. In clear contrast with equilibrium we find that the velocities are coupled to positions. Our model shows that, even for two particles only, the individual velocities display a variance depending on the interparticle separation and the emergence of correlations between the velocities of the particles. When considering systems composed of many particles we find an analytic expression connecting the overall velocity variance to density, at the mean-field level, and to the pair distribution function valid in the limit of small noise correlation times. Finally we discuss the intriguing analogies and main differences between our effective free energy functional and the theoretical scenario proposed so far for phase-separating active particles.
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Affiliation(s)
| | - Matteo Paoluzzi
- Department of Physics, Syracuse University, Syracuse, NY, USA
| | - Claudio Maggi
- Dipartimento di Fisica, Universita Sapienza, Rome, Italy
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Marini Bettolo Marconi U, Maggi C. Towards a statistical mechanical theory of active fluids. SOFT MATTER 2015; 11:8768-8781. [PMID: 26387914 DOI: 10.1039/c5sm01718a] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We present a stochastic description of a model of N mutually interacting active particles in the presence of external fields and characterize its steady state behavior in the absence of currents. To reproduce the effects of the experimentally observed persistence of the trajectories of the active particles we consider a Gaussian force having a non-vanishing correlation time τ, whose finiteness is a measure of the activity of the system. With these ingredients we show that it is possible to develop a statistical mechanical approach similar to the one employed in the study of equilibrium liquids and to obtain the explicit form of the many-particle distribution function by means of the multidimensional unified colored noise approximation. Such a distribution plays a role analogous to the Gibbs distribution in equilibrium statistical mechanics and provides complete information about the microscopic state of the system. From here we develop a method to determine the one- and two-particle distribution functions in the spirit of the Born-Green-Yvon (BGY) equations of equilibrium statistical mechanics. The resulting equations which contain extra-correlations induced by the activity allow us to determine the stationary density profiles in the presence of external fields, the pair correlations and the pressure of active fluids. In the low density regime we obtained the effective pair potential ϕ(r) acting between two isolated particles separated by a distance, r, showing the existence of an effective attraction between them induced by activity. Based on these results, in the second half of the paper we propose a mean field theory as an approach simpler than the BGY hierarchy and use it to derive a van der Waals expression of the equation of state.
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Affiliation(s)
- Umberto Marini Bettolo Marconi
- Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri, 62032, Camerino, INFN Perugia, Italy.
| | - Claudio Maggi
- Dipartimento di Fisica, Università di Roma Sapienza, I-00185, Rome, Italy
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Maggi C, Marconi UMB, Gnan N, Di Leonardo R. Multidimensional stationary probability distribution for interacting active particles. Sci Rep 2015; 5:10742. [PMID: 26021260 PMCID: PMC4448265 DOI: 10.1038/srep10742] [Citation(s) in RCA: 107] [Impact Index Per Article: 11.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2015] [Accepted: 04/29/2015] [Indexed: 11/13/2022] Open
Abstract
We derive the stationary probability distribution for a non-equilibrium system composed by an arbitrary number of degrees of freedom that are subject to Gaussian colored noise and a conservative potential. This is based on a multidimensional version of the Unified Colored Noise Approximation. By comparing theory with numerical simulations we demonstrate that the theoretical probability density quantitatively describes the accumulation of active particles around repulsive obstacles. In particular, for two particles with repulsive interactions, the probability of close contact decreases when one of the two particle is pinned. Moreover, in the case of isotropic confining potentials, the radial density profile shows a non trivial scaling with radius. Finally we show that the theory well approximates the “pressure” generated by the active particles allowing to derive an equation of state for a system of non-interacting colored noise-driven particles.
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Affiliation(s)
- Claudio Maggi
- Dipartimento di Fisica, Universit`a di Roma "Sapienza", Roma, I-00185, Italy
| | - Umberto Marini Bettolo Marconi
- Scuola di Scienze e Tecnologie, Universit`a di Camerino, Via Madonna delle Carceri, Camerino, INFN Perugia, 62032, Italy
| | - Nicoletta Gnan
- CNR-ISC, UOS Sapienza, P.le A. Moro2, Roma, I-00185, Italy
| | - Roberto Di Leonardo
- 1] Dipartimento di Fisica, Universit`a di Roma "Sapienza", Roma, I-00185, Italy [2] CNR-IMIP, UOS Roma, Dipartimento di Fisica Universit`a Sapienza, Roma, I-00185, Italy
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Wang J, Zhu S, Yin J. Saturation effects on intensity fluctuations of a laser with multiplicative white noise. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1995; 51:5035-5038. [PMID: 9912202 DOI: 10.1103/physreva.51.5035] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Cao L, Wu D, Luo X. First-order correction to the unified colored-noise approximation and dye-laser transients. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 50:4244-4252. [PMID: 9911398 DOI: 10.1103/physreva.50.4244] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Zhu S. Saturation effects in a two-mode ring laser. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 50:1710-1715. [PMID: 9911063 DOI: 10.1103/physreva.50.1710] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Li C, Da-jin W, Ling L. First-order-like transition for colored saturation models of dye lasers: Effects of quantum noise. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 49:506-516. [PMID: 9910256 DOI: 10.1103/physreva.49.506] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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