Particles with Internal Degrees of Freedom
Abstract
A development of a structurally complex particle concept to the case of general interaction potentials between its components is proposed. A structurally complex particle is defined by the restricted problem of N+1 bodies of different masses, which is reduced to the N body problem. Based on the Lagrangian description, a Hamiltonian formalism is formulated and the resulting first integrals of motion are discussed. The absence of equipartition among the degrees of freedom is demonstrated. Using the virial theorem, it is
proved that the energy of internal degrees of freedom does not exceed a certain fraction of the initial energy, depending on the ratio of the masses of internal particles to the mass of the shell. An exactly integrable case of a one-dimensional particle with two internal degrees of freedom is considered. The oscillation frequencies of such a structurally complex particle are determined by their masses and the mass of the shell. The energy distribution over them is not uniform. Universal relations are obtained for the distribution of the kinetic and potential energies of internal particles, which are independent of the choice of initial conditions. These relations provide
more detailed information about the energy distribution than the virial relations.
Downloads
References
V. V. Yanovsky, and A. V. Tur, Particles with internal degrees of freedom, Problems of Theoretical Physics, ScientificWorks, issue 2, (V. N. Karazin Kharkiv National University, 2016).
D.M. Naplekov, and V.V. Yanovsky, ”Distribution of energy in the ideal gas that lacks equipartition,” Scientific Reports, 13, 3427 (2023). https://doi.org/10.1038/s41598-023-30636-6
S.V. Slipushenko, A.V. Tur, and V.V. Yanovsky, ”Scattering of particles with internal degrees of freedom,” Journal of Experimental and Theoretical Physics, 117(2), 274-292 (2013). https://doi.org/10.1134/S1063776113080207
Ratner M. A., Tur A. V., and Yanovsky V. V., ”Peculiarities of nanoparticle reflection from a barrier,” Journal of Computational and Theoretical Nanoscience, 12, 589-594 (2015). https://doi.org/10.1166/jctn.2015.3771
B. W. Smith, M. Monthioux, and D. E. Luzzi, ”Encapsulated C60 in carbon nanotubes,” Nature, 396, 323-324 (1998). https://doi.org/10.1038/24521
M. Monthioux, ”Filling single-wall carbon nanotubes,” Carbon, 40(10), 1809-1823 (2002). https://doi.org/10.1016/S0008-6223(02)00102-1
M. Yudasaka, K. Ajima, K. Suenaga, T. Ichihashi, A. Hashimoto, and S. Iijima, ”Nano-extraction and nano-condensation for C60 incorporation into single-wall carbon nanotubes in liquid phases,” Chemical Phys. Lett. 380, 42-46 (2003). https://doi.org/10.1016/j.cplett.2003.08.095
S. Rols, J. Cambedouzou, M. Chorro, H. Schober, V. Agafonov, P. Launois, V. Davydov, et al., ”How Confinement Affects the Dynamics of C60 in Carbon Nanopeapods,” Phys. Rev. Lett. 101, 065507 (2008). https://doi.org/10.1103/PhysRevLett.101.065507
Gottfried Schill, Catenanes, rotaxanes and knots, (Academic Press, 1971).
I. T. Harrison, , and S. Harrison, ”Synthesis of a stable complex of a macrocycle and a threaded chain,” J. Am. Chem. Soc. 89(22), 5723-5724 (1967). https://doi.org/10.1021/ja00998a052
T.T. Perkins, D.E. Smith, R.G.Larson, and S.Chu, ”Single polymer dynamics in an elongational flow,” Science, 286, 83 (1995). https://doi.org/10.1126/science.276.5321.2016
S.R. Quake, H. Babcock, and S. Chu, ”The dynamics of partially extended single molecules of DNA,” Nature, 388(6638), 151-154 (1997). https://doi.org/10.1038/40588
D. Wirtz, ”Direct Measurement of the Transport Properties of a Single DNA Molecule,” Phys. Rev. Lett. 75, 2436 (1995). https://doi.org/10.1103/PhysRevLett.75.2436
A. Blumen, ”Anomalous diffusion of particles with internal degrees of freedom,” Philosophical Magazine Part B, 81(9), 1021-1031 (2001). http://dx.doi.org/10.1080/13642810108205788
P.E. Rouse, Jr., ”A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers,” J. Chem. Phys. 21, 1272-1280 (1953). https://doi.org/10.1063/1.1699180
P.J. Flory, Statistical Mechanics of Chain Molecules, (Interscience, 1969).
M.R. Buche, and M. N. Silberstein, ”Chain breaking in the statistical mechanical constitutive theory of polymer networks,” Journal of the Mechanics and Physics of Solids, 156, 104593 (2021). https://doi.org/10.1016/j.jmps.2021.104593
N. Balabaev, and T.Khazanovich, ”Extension of chains composed of freely joined elastic segments,” Russian Journal of Physical Chemistry B, 3, 242 (2009).
F. Manca, S. Giordano, P. L. Palla, R. Zucca, F. Cleri, and L. Colombo, ”Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles,” The Journal of Chemical Physics, 136, 154906 (2012). https://doi.org/10.1063/1.4704607
M. Schliwa, and G. Woehlke, ”Molecular motors, Nature,” 422, 759-765 (2003). https://doi.org/10.1038/nature01601
H. C. Berg, and R. A. Anderson, ”Bacteria swim by rotating their flagellar filaments,” Nature, 245, 380-382 (1973). https://doi.org/10.1038/245380a0
Z. Sadjadi, M. R. Shaebani, H. Rieger, and L. Santen, ”Persistent- random- walk approach to anomalous transport of self- propelled particles,” Phys. Rev. E, 91, 062715 (2015). https://doi.org/10.1103/PhysRevE.91.062715
D. Chowdhury, ”Stochastic mechano- chemical kinetics of molecular motors: a multidisciplinary enterprise from a physicist’s perspective,” Phys. Rep. 529, 1-197, (2013). https://doi.org/10.1016/j.physrep.2013.03.005
Active Matter, edited by S. Tibbits, (The MIT Press, 2017). https://doi.org/10.7551/mitpress/11236.001.0001
V.I. Arnold, Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60, 2nd ed., (Springer, 1989). https://doi.org/10.1007/978-1-4757-2063-1.
J.R. Taylor, Title Classical Mechanics, (University Science Books, 2005).
E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, (Cambridge University Press, 1999).
Copyright (c) 2026 Kostyantyn M. Kulyk, Maryna A. Ratner, Volodymyr V. Yanovsky
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
