{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:51Z","timestamp":1760145711605,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,25]],"date-time":"2024-08-25T00:00:00Z","timestamp":1724544000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Foundation of State Key Laboratory of Laser Interaction","award":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"],"award-info":[{"award-number":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"]}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"],"award-info":[{"award-number":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Outstanding Youth Fund of Hunan Province","award":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"],"award-info":[{"award-number":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"],"award-info":[{"award-number":["SKLLIM2107","SKLLIM1907","SKLLIM2205","12375234","2022JJ20051","2023RC44"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The non-equilibrium characteristics during the shock relaxation process hold a foundational position in various fields. In contrast to the propagation of a single shock wave, the collision process of two shock waves exhibits distinct non-equilibrium features. Employing non-equilibrium molecular dynamics, we simulated the collision of ultra-strong shock waves in a classical gas system, investigating the relationship between equilibrium relaxation time and shock intensity. Tracking the spatial migration of microscopic particles in the shock collision region during the relaxation process, we observed a significant contribution of particle migration to the average energy changes during relaxation. The discussion on particle migration provides a valuable new perspective for understanding the microscopic mechanisms of the relaxation process.<\/jats:p>","DOI":"10.3390\/e26090724","type":"journal-article","created":{"date-parts":[[2024,8,26]],"date-time":"2024-08-26T03:59:09Z","timestamp":1724644749000},"page":"724","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Effects of Particle Migration on the Relaxation of Shock Wave Collisions"],"prefix":"10.3390","volume":"26","author":[{"given":"Hao","family":"Li","sequence":"first","affiliation":[{"name":"Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-6673-7064","authenticated-orcid":false,"given":"Bo","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9076-3671","authenticated-orcid":false,"given":"Zixiang","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8716-3213","authenticated-orcid":false,"given":"Xinyu","family":"Zhang","sequence":"additional","affiliation":[{"name":"Center for Applied Physics and Technology, and College of Engineering, Peking University, Beijing 100871, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chongjie","family":"Mo","sequence":"additional","affiliation":[{"name":"Beijing Computational Science Research Center, Beijing 100193, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Quanxi","family":"Xue","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Laser Interaction with Matter, Northwest Institute of Nuclear Technology, Xi\u2019an 710024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiazi","family":"Xiao","sequence":"additional","affiliation":[{"name":"Department of Mechanics, School of Civil Engineering, Central South University, Changsha 410075, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7405-1578","authenticated-orcid":false,"given":"Hao","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"062705","DOI":"10.1063\/1.1927099","article-title":"Shock propagation in deuterium-tritium-saturated foam","volume":"12","author":"Collins","year":"2005","journal-title":"Phys. Plasmas"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"195001","DOI":"10.1103\/PhysRevLett.129.195001","article-title":"Shock-Augmented Ignition Approach to Laser Inertial Fusion","volume":"129","author":"Scott","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"018401","DOI":"10.1063\/1.5131748","article-title":"Understanding the effects of radiative preheat and self-emission from shock heating on equation of state measurement at 100s of Mbar using spherically converging shock waves in a NIF hohlraum","volume":"5","author":"Nilsen","year":"2020","journal-title":"Matter Radiat. Extrem."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"195001","DOI":"10.1103\/PhysRevLett.119.195001","article-title":"Electron Shock Ignition of Inertial Fusion Targets","volume":"119","author":"Shang","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1093\/mnras\/182.2.147","article-title":"The acceleration of cosmic rays in shock fronts","volume":"182","author":"Bell","year":"1978","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215002","DOI":"10.1103\/PhysRevLett.107.215002","article-title":"Electron Temperature Gradient Scale at Collisionless Shocks","volume":"107","author":"Schwartz","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kyrala, G. (2005). Laboratory Simulations of Supernova Shockwave Propagation. High Energy Density Laboratory Astrophysics, Springer.","DOI":"10.1007\/1-4020-4162-4"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s11467-013-0317-9","article-title":"The Physics and Astrophysics of Type Ia Supernova Explosions","volume":"8","author":"Guidry","year":"2013","journal-title":"Front. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"125103","DOI":"10.1063\/1.3664124","article-title":"Effect of shocklets on the velocity gradients in highly compressible isotropic turbulence","volume":"23","author":"Wang","year":"2011","journal-title":"Phys. Fluids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1017\/jfm.2012.474","article-title":"Effect of compressibility on the small-scale structures in isotropic turbulence","volume":"713","author":"Wang","year":"2012","journal-title":"J. Fluid Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1792","DOI":"10.1063\/1.857960","article-title":"Shock wave effects on a turbulent flow","volume":"3","author":"Rotman","year":"1991","journal-title":"Phys. Fluids A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"124503","DOI":"10.1063\/1.1860554","article-title":"Shock wave propagation in dissociating low-Z liquids: D2","volume":"122","author":"Belonoshko","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"064120","DOI":"10.1103\/PhysRevB.72.064120","article-title":"Atomistic simulations of shock-induced transformations and their orientation dependence in bcc Fe single crystals","volume":"72","author":"Kadau","year":"2005","journal-title":"Phys. Rev. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2174","DOI":"10.1103\/PhysRevLett.70.2174","article-title":"Detonations at nanometer resolution using molecular dynamics","volume":"70","author":"Brenner","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1146\/annurev-matsci-080819-120123","article-title":"Chemistry Under Shock Conditions","volume":"51","author":"Hamilton","year":"2021","journal-title":"Annu. Rev. Mater. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"155001","DOI":"10.1103\/PhysRevLett.98.155001","article-title":"Shock Ignition of Thermonuclear Fuel with High Areal Density","volume":"98","author":"Betti","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"022024","DOI":"10.1088\/1742-6596\/112\/2\/022024","article-title":"Shock ignition of thermonuclear fuel with high areal densities","volume":"112","author":"Betti","year":"2008","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"045004","DOI":"10.1103\/PhysRevLett.103.045004","article-title":"Shock Ignition: A New Approach to High Gain Inertial Confinement Fusion on the National Ignition Facility","volume":"103","author":"Perkins","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"20200015","DOI":"10.1098\/rsta.2020.0015","article-title":"Double-cone ignition scheme for inertial confinement fusion","volume":"378","author":"Zhang","year":"2020","journal-title":"Philos. Trans. R. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"189","DOI":"10.4208\/cicp.OA-2018-0050","article-title":"A Hybrid Numerical Simulation of Supersonic Isotropic Turbulence","volume":"25","author":"Liu","year":"2019","journal-title":"Commun. Comput. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1103\/PhysRev.82.885","article-title":"The Solution of the Boltzmann Equation for a Shock Wave","volume":"82","year":"1951","journal-title":"Phys. Rev."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.1063\/1.1706894","article-title":"Bimodal Distributions and Plasma Shock Wave Structure","volume":"6","author":"Comisar","year":"1963","journal-title":"Phys. Fluids"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1063\/1.1706528","article-title":"Some Aspects of Shock Structure According to the Bimodal Model","volume":"5","author":"Muckenfuss","year":"1962","journal-title":"Phys. Fluids"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1134\/1.567510","article-title":"Shock wave structure in dense gases","volume":"66","author":"Zhakhovskii","year":"1997","journal-title":"JETP Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/BF01435522","article-title":"Atomistic computer simulations of shock waves","volume":"5","author":"Holian","year":"1995","journal-title":"Shock Waves"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2798","DOI":"10.1103\/PhysRevA.22.2798","article-title":"Shock-wave structure via nonequilibrium molecular dynamics and Navier-Stokes continuum mechanics","volume":"22","author":"Holian","year":"1980","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s00162-023-00683-w","article-title":"Theory and simulation of shock waves freely propagating through monoatomic non-Boltzmann gas","volume":"38","year":"2024","journal-title":"Theor. Comput. Fluid Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"144504","DOI":"10.1103\/PhysRevLett.112.144504","article-title":"Shock-Wave Compression and Joule-Thomson Expansion","volume":"112","author":"Hoover","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1531","DOI":"10.1103\/PhysRevLett.42.1531","article-title":"Structure of a Shock-Wave Front in a Liquid","volume":"42","author":"Hoover","year":"1979","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Hoover, W.G., and Hoover, C.G. (2010). Shockwaves and Local Hydrodynamics: Failure of the Navier-Stokes Equations. New Trends in Statistical Physics, World Scientific.","DOI":"10.1142\/9789814307543_0002"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"026707","DOI":"10.1103\/PhysRevE.82.026707","article-title":"Heat-flow equation motivated by the ideal-gas shock wave","volume":"82","author":"Holian","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"114502","DOI":"10.1063\/1.3486088","article-title":"Test of a new heat-flow equation for dense-fluid shock waves","volume":"133","author":"Holian","year":"2010","journal-title":"J. Chem Phy."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"R24","DOI":"10.1103\/PhysRevE.47.R24","article-title":"Modeling shock waves in an ideal gas: Going beyond the Navier-Stokes level","volume":"47","author":"Holian","year":"1993","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1103\/PhysRevLett.83.1175","article-title":"Shock Wave Structure in Lennard-Jones Crystal via Molecular Dynamics","volume":"83","author":"Zybin","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2085","DOI":"10.1126\/science.280.5372.2085","article-title":"Plasticity induced by shock waves in nonequilibrium molecular-dynamics simulations","volume":"280","author":"Holian","year":"1998","journal-title":"Science"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"125505","DOI":"10.1103\/PhysRevLett.109.125505","article-title":"Evolution of Shock-Induced Orientation-Dependent Metastable States in Crystalline Aluminum","volume":"109","author":"Budzevich","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"070003","DOI":"10.1063\/1.4971591","article-title":"MD simulation of steady shock-wave fronts with phase transition in single-crystal iron","volume":"1793","author":"Zhakhovsky","year":"2017","journal-title":"AIP Conf. Proc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"144105","DOI":"10.1103\/PhysRevB.108.144105","article-title":"Shock-induced volume-collapse phase transition in a Ce-La alloy dynamically compressed up to 20 GPa","volume":"108","author":"Gu","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"023201","DOI":"10.1103\/PhysRevE.95.023201","article-title":"Molecular dynamics simulation of strong shock waves propagating in dense deuterium, taking into consideration effects of excited electrons","volume":"95","author":"Liu","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"023207","DOI":"10.1103\/PhysRevE.102.023207","article-title":"Dynamics of bond breaking and formation in polyethylene near shock front","volume":"102","author":"Liu","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"108171","DOI":"10.1016\/j.cpc.2021.108171","article-title":"LAMMPS\u2014A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales","volume":"271","author":"Thompson","year":"2022","journal-title":"Computer. Phys. Commun."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1103\/PhysRevLett.53.1248","article-title":"Shock Compression of Liquid Helium to 56 GPa (560 kbar)","volume":"53","author":"Nellis","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1063\/1.447334","article-title":"A unified formulation of the constant temperature molecular dynamics methods","volume":"81","year":"1984","journal-title":"J. Chem. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1103\/PhysRevA.31.1695","article-title":"Canonical dynamics: Equilibrium phase-space distributions","volume":"31","author":"Hoover","year":"1985","journal-title":"Phys. Rev. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"134103","DOI":"10.1103\/PhysRevB.69.134103","article-title":"Rapid estimation of elastic constants by molecular dynamics simulation under constant stress","volume":"69","author":"Shinoda","year":"2004","journal-title":"Phys. Rev. B"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.molstruc.2014.04.006","article-title":"Analytical treatment of second virial coefficient over Lennard-Jones (2n-n) potential and its application to molecular systems","volume":"1068","author":"Mamedov","year":"2014","journal-title":"J. Mol. Struct."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1080\/00268976800100381","article-title":"A new interatomic potential function for helium","volume":"14","author":"Beck","year":"1968","journal-title":"Mol. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"020002","DOI":"10.1063\/1.5122325","article-title":"Influence of defects on the diffusion of helium in uranium dioxide: Molecular dynamics study","volume":"2142","author":"Kovalenko","year":"2019","journal-title":"AIP Conf. Proc."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Liu, H., Kang, W., Zhang, Q., Zhang, Y., Duan, H., and He, X. (2016). Molecular dynamics simulations of microscopic structure of ultra strong shock waves in dense helium. Front. Phys., 11.","DOI":"10.1007\/s11467-016-0590-5"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1063\/1.1747782","article-title":"The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics","volume":"18","author":"Irving","year":"1950","journal-title":"J. Chem. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1038\/072294b0","article-title":"The problem of the random walk","volume":"72","author":"Pearson","year":"1905","journal-title":"Nature"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/9\/724\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:42:32Z","timestamp":1760110952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/9\/724"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,25]]},"references-count":51,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["e26090724"],"URL":"https:\/\/doi.org\/10.3390\/e26090724","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,8,25]]}}}