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Published: 18 May 2024

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1

Ford, G. W., J. T. Lewis, and R. F. O’Connell. "Quantum Langevin equation." Physical Review A 37, no. 11 (June 1, 1988): 4419–28. http://dx.doi.org/10.1103/physreva.37.4419.

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2

de Oliveira, Mário J. "Quantum Langevin equation." Journal of Statistical Mechanics: Theory and Experiment 2020, no. 2 (February 21, 2020): 023106. http://dx.doi.org/10.1088/1742-5468/ab6de2.

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3

Pomeau, Yves, and Jarosław Piasecki. "The Langevin equation." Comptes Rendus Physique 18, no. 9-10 (November 2017): 570–82. http://dx.doi.org/10.1016/j.crhy.2017.10.001.

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4

Wu, Xiongwu, Bernard R. Brooks, and Eric Vanden-Eijnden. "Self-guided Langevin dynamics via generalized Langevin equation." Journal of Computational Chemistry 37, no. 6 (July 16, 2015): 595–601. http://dx.doi.org/10.1002/jcc.24015.

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5

Sekimoto, Ken. "Langevin Equation and Thermodynamics." Progress of Theoretical Physics Supplement 130 (1998): 17–27. http://dx.doi.org/10.1143/ptps.130.17.

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6

Jaekel, M. T. "Stochastic quantum Langevin equation." Journal of Physics A: Mathematical and General 22, no. 5 (March 7, 1989): 537–57. http://dx.doi.org/10.1088/0305-4470/22/5/017.

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7

Gillespie, Daniel T. "The chemical Langevin equation." Journal of Chemical Physics 113, no. 1 (July 2000): 297–306. http://dx.doi.org/10.1063/1.481811.

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8

Savović, Svetislav, Linqing Li, Isidora Savović, Alexandar Djordjevich, and Rui Min. "Treatment of Mode Coupling in Step-Index Multimode Microstructured Polymer Optical Fibers by the Langevin Equation." Polymers 14, no. 6 (March 19, 2022): 1243. http://dx.doi.org/10.3390/polym14061243.

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By solving the Langevin equation, mode coupling in a multimode step-index microstructured polymer optical fibers (SI mPOF) with a solid core was investigated. The numerical integration of the Langevin equation was based on the computer-simulated Langevin force. The numerical solution of the Langevin equation corresponded to the previously reported theoretical data. We demonstrated that by solving the Langevin equation (stochastic differential equation), one can successfully treat a mode coupling in multimode SI mPOF as a stochastic process, since it is caused by its intrinsic random perturbations. Thus, the Langevin equation allowed for a stochastic mathematical description of mode coupling in SI mPOF. Regarding the efficiency and execution speed, the Langevin equation was more favorable than the power flow equation. Such knowledge is useful for the use of multimode SI mPOFs for potential sensing and communication applications.
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9

Ahmad, Bashir, and Juan J. Nieto. "Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions." International Journal of Differential Equations 2010 (2010): 1–10. http://dx.doi.org/10.1155/2010/649486.

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We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.
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10

Khalili Golmankhaneh, Alireza. "On the Fractal Langevin Equation." Fractal and Fractional 3, no. 1 (March 13, 2019): 11. http://dx.doi.org/10.3390/fractalfract3010011.

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In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- τ Cantor set. The fractal mean square displacement of different random walks on the middle- τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.
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11

SATIN, SEEMA, and A. D. GANGAL. "LANGEVIN EQUATION ON FRACTAL CURVES." Fractals 24, no. 03 (August 30, 2016): 1650028. http://dx.doi.org/10.1142/s0218348x16500286.

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We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the F[Formula: see text]-Calculus.
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12

Chvosta, Petr, and Frantisek Slanina. "Langevin equation with back-reaction." Journal of Physics A: Mathematical and General 35, no. 21 (May 18, 2002): L277—L282. http://dx.doi.org/10.1088/0305-4470/35/21/101.

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13

Shiktorov, P., E. Starikov, V. Gru inskis, and L. Reggiani. "A macroscopic quantum Langevin equation." Semiconductor Science and Technology 19, no. 4 (March 8, 2004): S232—S234. http://dx.doi.org/10.1088/0268-1242/19/4/078.

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14

Tanaka, Satoshi. "General Langevin Equation and Anomaly." Progress of Theoretical Physics Supplement 111 (1993): 263–74. http://dx.doi.org/10.1143/ptps.111.263.

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15

Hayakawa, Hisao. "Langevin equation with Coulomb friction." Physica D: Nonlinear Phenomena 205, no. 1-4 (June 2005): 48–56. http://dx.doi.org/10.1016/j.physd.2004.12.011.

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16

Schnoerr, David, Guido Sanguinetti, and Ramon Grima. "The complex chemical Langevin equation." Journal of Chemical Physics 141, no. 2 (July 14, 2014): 024103. http://dx.doi.org/10.1063/1.4885345.

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17

Antoniou, Dimitri, and Steven D. Schwartz. "Langevin equation in momentum space." Journal of Chemical Physics 119, no. 21 (December 2003): 11329–34. http://dx.doi.org/10.1063/1.1623183.

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18

Migdal, A. A., M. A. Bershadskii, and T. A. Kozhamkulov. "Langevin equation in field theory." Soviet Physics Journal 29, no. 3 (March 1986): 211–20. http://dx.doi.org/10.1007/bf00891882.

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19

Söderberg, B. "On the complex Langevin equation." Nuclear Physics B 295, no. 3 (March 1988): 396–408. http://dx.doi.org/10.1016/0550-3213(88)90361-6.

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20

Ford, G. W., and M. Kac. "On the quantum langevin equation." Journal of Statistical Physics 46, no. 5-6 (March 1987): 803–10. http://dx.doi.org/10.1007/bf01011142.

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21

Bargueño, Pedro, and Salvador Miret-Artés. "The generalized Schrödinger–Langevin equation." Annals of Physics 346 (July 2014): 59–65. http://dx.doi.org/10.1016/j.aop.2014.04.004.

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22

Mendoza-Suárez, Jairo Alonso, Juan Carlos López-Carreño, and Rosalba Mendoza-Suárez. "Another Solution to the Schrödinger-Langevin Equation." Revista Lasallista de Investigación 18, no. 1 (October 20, 2021): 25–33. http://dx.doi.org/10.22507/rli.v18n1a2.

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Introduction: an alternative solution to the Schrodinger-Langevin equation is presented, where the temporal dependence is explained, assuming a Coulomb potential. Finally, the trajectory equations are found. Objective: in this paper we contribute by presenting a detailed and simple solution of the Schrödinger-Langevin equation for a Coulomb potential. Materials and Methods: using an appropriate ansatz, we solve the Schrödinger-Langevin equation, finding the expected values of position and moment. Results: a simple method was presented to find the expected position and moment values in the Schrödinger-Langevin equation, the ansatz used to find these solutions allows the model to be generalized in a certain way to electric potentials and harmonic oscillators. Conclusions: the model used to solve the Schrödinger-Langevin equation, allowed to find the expected values of position and moment of a particle in a Coulomb potential, the temporal dependence of such solutions is made explicit, which allows finding the path equations of the particles.
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23

Schienbein, M., and H. Gruler. "Langevin equation, Fokker-Planck equation and cell migration." Bulletin of Mathematical Biology 55, no. 3 (May 1993): 585–608. http://dx.doi.org/10.1007/bf02460652.

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24

SCHIENBEIN, M., and H. GRULER. "Langevin equation, Fokker-Planck equation and cell migration." Bulletin of Mathematical Biology 55, no. 3 (May 1993): 585–608. http://dx.doi.org/10.1016/s0092-8240(05)80241-1.

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25

Zhao, Kaihong. "Existence and uh-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag-Leffler functions." Filomat 37, no. 4 (2023): 1053–63. http://dx.doi.org/10.2298/fil2304053z.

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The Langevin equation is a very important mathematical model in describing the random motion of particles. The fractional Langevin equation is a powerful tool in complex viscoelasticity. Therefore, this paper focuses on a class of nonlinear higher-order Hadamard fractional Langevin equation with integral boundary value conditions. Firstly, we employ successive approximation and Mittag-Leffler function to transform the differential equation into an equivalent integral equation. Then the existence and uniqueness of the solution are obtained by using the fixed point theory. Meanwhile, the Ulam-Hyers (UH) stability is proved by inequality technique and direct analysis.
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26

Guo, Peng, Changpin Li, and Fanhai Zeng. "Numerical simulation of the fractional Langevin equation." Thermal Science 16, no. 2 (2012): 357–63. http://dx.doi.org/10.2298/tsci110407073g.

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In this paper, we study the fractional Langevin equation, whose derivative is in Caputo sense. By using the derived numerical algorithm, we obtain the displacement and the mean square displacement which describe the dynamic behaviors of the fractional Langevin equation.
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27

Zhao, Kaihong. "Existence, Stability and Simulation of a Class of Nonlinear Fractional Langevin Equations Involving Nonsingular Mittag–Leffler Kernel." Fractal and Fractional 6, no. 9 (August 26, 2022): 469. http://dx.doi.org/10.3390/fractalfract6090469.

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The fractional Langevin equation is a very effective mathematical model for depicting the random motion of particles in complex viscous elastic liquids. This manuscript is mainly concerned with a class of nonlinear fractional Langevin equations involving nonsingular Mittag–Leffler (ML) kernel. We first investigate the existence and uniqueness of the solution by employing some fixed-point theorems. Then, we apply direct analysis to obtain the Ulam–Hyers (UH) type stability. Finally, the theoretical analysis and numerical simulation of some interesting examples show that there is a great difference between the fractional Langevin equation and integer Langevin equation in describing the random motion of free particles.
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28

MOCHIZUKI, RIUJI. "STOCHASTIC CALCULUS AND COVARIANT AND ROTATION-INVARIANT LANGEVIN EQUATION FOR GRAVITY." Modern Physics Letters A 05, no. 28 (November 10, 1990): 2335–42. http://dx.doi.org/10.1142/s0217732390002687.

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We investigate the relations among the Langevin equation, the Fokker-Planck equation, and the stochastic action, both in the sense of Ito and of Stratonovich. In the latter case we suggest a somewhat modified Langevin equation which is covariant and rotation-invariant.
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29

Zhao, Kaihong. "Stability of a Nonlinear ML-Nonsingular Kernel Fractional Langevin System with Distributed Lags and Integral Control." Axioms 11, no. 7 (July 21, 2022): 350. http://dx.doi.org/10.3390/axioms11070350.

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The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation in portraying Brownian motion. This paper focuses on a nonlinear ML-fractional Langevin system with distributed lag and integral control. Employing the fixed-point theorem of generalised metric space established by Diaz and Margolis, we built the Hyers–Ulam–Rassias (HUR) stability along with Hyers–Ulam (HU) stability of this ML-fractional Langevin system. Applying our main results and MATLAB software, we have carried out theoretical analysis and numerical simulation on an example. By comparing with the numerical simulation of the corresponding classical Langevin system, it can be seen that the ML-fractional Langevin system can better reflect the stationarity of random particles in the statistical sense.
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30

GROTHAUS, MARTIN, and PATRIK STILGENBAUER. "GEOMETRIC LANGEVIN EQUATIONS ON SUBMANIFOLDS AND APPLICATIONS TO THE STOCHASTIC MELT-SPINNING PROCESS OF NONWOVENS AND BIOLOGY." Stochastics and Dynamics 13, no. 04 (October 7, 2013): 1350001. http://dx.doi.org/10.1142/s0219493713500019.

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In this paper we develop geometric versions of the classical Langevin equation on regular submanifolds in Euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometric Langevin equation has already been detected before by Lelièvre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant Euclidean norm. The latters are seemingly new and provide a galaxy of new, beautiful and powerful mathematical models. Up to the authors best knowledge there are not many mathematical papers available dealing with geometric Langevin processes. We connect the first version of the geometric Langevin equation via proving that its generator coincides with the generalized Langevin operator proposed by Soloveitchik, Jørgensen or Kolokoltsov. All our studies are strongly motivated by industrial applications in modeling the fiber lay-down dynamics in the production process of nonwovens. We light up the geometry occurring in these models and show up the connection with the spherical velocity version of the geometric Langevin process. Moreover, as a main point, we construct new smooth industrial relevant three-dimensional fiber lay-down models involving the spherical Langevin process. Finally, relations to a class of swarming models are presented and further applications of the geometric Langevin equations are given.
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31

Chauleur, Quentin. "Dynamics of the Schrödinger–Langevin equation." Nonlinearity 34, no. 4 (February 18, 2021): 1943–74. http://dx.doi.org/10.1088/1361-6544/abd528.

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32

Xu, Mengrui, Shurong Sun, and Zhenlai Han. "SOLVABILITY FOR IMPULSIVE FRACTIONAL LANGEVIN EQUATION." Journal of Applied Analysis & Computation 10, no. 2 (2020): 486–94. http://dx.doi.org/10.11948/20180170.

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33

Garrido, P. L., F. de los Santos, and M. A. Muñoz. "Langevin equation for driven diffusive systems." Physical Review E 57, no. 1 (January 1, 1998): 752–55. http://dx.doi.org/10.1103/physreve.57.752.

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34

Lee, M. Howard. "Generalized Langevin equation and recurrence relations." Physical Review E 62, no. 2 (August 1, 2000): 1769–72. http://dx.doi.org/10.1103/physreve.62.1769.

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35

Lubashevsky, Ihor. "Towards Multi-Dimensional Nonlinear Langevin Equation." Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications 2014 (May 5, 2014): 278–83. http://dx.doi.org/10.5687/sss.2014.278.

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36

Altaisky, M. V. "Langevin equation with scale-dependent noise." Doklady Physics 48, no. 9 (September 2003): 478–80. http://dx.doi.org/10.1134/1.1616054.

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37

Chakrabarti, R., and R. Vasudevan. "Quantum Langevin equation: a quadratic system." Journal of Physics A: Mathematical and General 23, no. 14 (July 21, 1990): 3215–26. http://dx.doi.org/10.1088/0305-4470/23/14/019.

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38

Martı́n, Rosario, and Enric Verdaguer. "On the semiclassical Einstein-Langevin equation." Physics Letters B 465, no. 1-4 (October 1999): 113–18. http://dx.doi.org/10.1016/s0370-2693(99)01068-0.

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39

Kemeny, G., S. D. Mahanti, and T. A. Kaplan. "Generalized Langevin equation for an oscillator." Physical Review B 34, no. 9 (November 1, 1986): 6288–94. http://dx.doi.org/10.1103/physrevb.34.6288.

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40

Lim, S. C., Ming Li, and L. P. Teo. "Langevin equation with two fractional orders." Physics Letters A 372, no. 42 (October 2008): 6309–20. http://dx.doi.org/10.1016/j.physleta.2008.08.045.

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41

van Kampen, N. G. "Derivation of the quantum Langevin equation." Journal of Molecular Liquids 71, no. 2-3 (April 1997): 97–105. http://dx.doi.org/10.1016/s0167-7322(97)00002-0.

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42

Li, Tao. "Chemical Langevin Equation for Complex Reactions." Journal of Physical Chemistry A 124, no. 5 (January 15, 2020): 810–16. http://dx.doi.org/10.1021/acs.jpca.9b10108.

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43

McPhie, M. G., P. J. Daivis, I. K. Snook, J. Ennis, and D. J. Evans. "Generalized Langevin equation for nonequilibrium systems." Physica A: Statistical Mechanics and its Applications 299, no. 3-4 (October 2001): 412–26. http://dx.doi.org/10.1016/s0378-4371(01)00328-4.

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44

Darzi, Rahmat, Bahram Agheli, and Juan J. Nieto. "Langevin Equation Involving Three Fractional Orders." Journal of Statistical Physics 178, no. 4 (January 2, 2020): 986–95. http://dx.doi.org/10.1007/s10955-019-02476-0.

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45

Zhang, Shu-dong, Qin-liang Fan, and E.-jiang Ding. "Critical processes, Langevin equation and universality." Physics Letters A 203, no. 2-3 (July 1995): 83–87. http://dx.doi.org/10.1016/0375-9601(95)00397-l.

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46

Constable, Steve, Matthew Schmidt, Christopher Ing, Tao Zeng, and Pierre-Nicholas Roy. "Langevin Equation Path Integral Ground State." Journal of Physical Chemistry A 117, no. 32 (June 19, 2013): 7461–67. http://dx.doi.org/10.1021/jp4015178.

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47

Toxvaerd, So/ren. "Solution of the generalized Langevin equation." Journal of Chemical Physics 82, no. 12 (June 15, 1985): 5658–62. http://dx.doi.org/10.1063/1.448552.

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48

van Kampen, N. G. "Langevin-like equation with colored noise." Journal of Statistical Physics 54, no. 5-6 (March 1989): 1289–308. http://dx.doi.org/10.1007/bf01044716.

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49

Shimizu, Toshihiro. "Generalized Langevin equation with chaotic force." Physica A: Statistical Mechanics and its Applications 212, no. 1-2 (December 1994): 61–74. http://dx.doi.org/10.1016/0378-4371(94)90137-6.

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50

Hamber, Herbert W., and Hai-cang Ren. "Complex probabilities and the Langevin equation." Physics Letters B 159, no. 4-6 (September 1985): 330–34. http://dx.doi.org/10.1016/0370-2693(85)90261-8.

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