Skip to main content
SpringerLink
Log in

Computational and Technical Aspects

  • Chapter
  • First Online:
Statistical Mechanics of Hamiltonian Systems with Bounded Kinetic Terms

Part of the book series: Springer Theses ((Springer Theses))

  • 227 Accesses

Abstract

Many of the results discussed in the previous Chapters are based on numerical simulations and data-processing methods. This Chapter is devoted to a careful analysis of the techniques employed throughout this Thesis, both from a mathematical and from an applicative point of view.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ammenti A, Cecconi F, Marini Bettolo Marconi U, Vulpiani A (2009) A statistical model for translocation of structured polypeptide chains through nanopores. J Phys Chem B 113:10348–10356

    Google Scholar 

  2. Ansalone P, Chinappi M, Rondoni L, Cecconi F (2015) Driven diffusion against electrostatic or effective energy barrier across \(\alpha \)-hemolysin. J Chem Phys 143:154109

    Google Scholar 

  3. Baldassarri A, Puglisi A, Sarracino A (2015) Coarsening in granular systems. C R Physique 16:291

    Google Scholar 

  4. Baldovin M, Cecconi F, Cencini M, Puglisi A, Vulpiani A (2018) The role of data in model building and prediction: a survey through examples. Entropy 20:807

    Google Scholar 

  5. Baldovin M, Cecconi F, Vulpiani A (2020) Effective equations for reaction coordinates in polymer transport. J Stat Mech Theory Exp 2020:013208

    Google Scholar 

  6. Baldovin M, Puglisi A, Vulpiani A (2018) Langevin equation in systems with also negative temperatures. J Stat Mech 2018:043207

    Google Scholar 

  7. Baldovin M, Puglisi A, Vulpiani A (2019) Langevin equations from experimental data: the case of rotational diffusion in granular media. PLOS ONE 14:1–16

    Article  Google Scholar 

  8. Benettin G, Giorgilli A (1994) On the Hamiltonian interpolation of nearto- the identity symplectic mappings with application to symplectic integration algorithms. J Stat Phys 74:1117–1143

    Article  ADS  Google Scholar 

  9. Bezrukov SM, Vodyanoy I, Parsegian VA (1994) Counting polymers moving through a single ion channel. Nature 370:279

    Google Scholar 

  10. Brunton SL, Proctor JL, Kutz JN (2016) Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc Natl Acad Sci U.S.A 113:3932–3937

    Article  ADS  MathSciNet  Google Scholar 

  11. Cavagna A (2009) Supercooled liquids for pedestrians. Phys Rep 476:51

    Google Scholar 

  12. Cressiot B, Oukhaled A, Patriarche G, Pastoriza-Gallego M, Betton J-M, Auvray L, Muthukumar M, Bacri L, Pelta J (2012) Protein transport through a narrow solid-state nanopore at high voltage: experiments and theory. ACS Nano 6:6236–6243

    Article  Google Scholar 

  13. D’Anna G, Mayor P, Barrat A, Loreto V, Nori F (2003) Observing Brownian motion in vibration-fluidized granular matter. Nature 424:909

    Google Scholar 

  14. Dyre JC (2006) The glass transition and elastic models of glass-forming liquids. Rev Mod Phys 78:953

    Google Scholar 

  15. Fiege A, Aspelmeier T, Zippelius A (2009) Long-time tails and cage effect in driven granular fluids. Phys Rev Lett 102:098001

    Google Scholar 

  16. Flyvbjerg H (1998) Error estimates on averages of correlated data. In: Kertész J, Kondor I (eds) Advances in computer simulation. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 88–103

    Google Scholar 

  17. Ford GW, Kac M, Mazur P (1965) Statistical mechanics of assemblies of coupled oscillators. J Math Phys 6:504–515

    Article  ADS  MathSciNet  Google Scholar 

  18. Frenkel D, Smit B (2001) Understanding molecular simulation: from algorithms to applications, vol 1. Elsevier

    Google Scholar 

  19. Friedrich R, Peinke J, Sahimi M, Reza Rahimi Tabar M (2011) Approaching complexity by stochastic methods: from biological systems to turbulence. Phys Rep 506:87–162

    Google Scholar 

  20. Gardiner CW (1985) Handbook of stochastic methods. Springer, Berlin

    Google Scholar 

  21. Gnoli A, Puglisi A, Touchette H (2013) Granular Brownian motion with dry friction. Europhys. Lett 102:14002

    Article  ADS  Google Scholar 

  22. Grabert H, Hänggi P, Talkner P (1980) Microdynamics and nonlinear stochastic processes of gross variables. J Stat Phys 22:537–552

    Article  ADS  MathSciNet  Google Scholar 

  23. Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. vol 31. Springer Science & Business Media

    Google Scholar 

  24. Hänggi P, Talkner P, Borkovec M (1990) Reaction-rate theory: fifty years after Kramers. Rev Mod Phys 62:251–341

    Article  ADS  MathSciNet  Google Scholar 

  25. Heussinger C, Berthier L, Barrat J-L (2010) Superdiffusive, heterogeneous, and collective particle motion near the fluid-solid transition in athermal disordered materials. Europhys Lett 90:20005

    Google Scholar 

  26. Honeycutt RL (1992) Stochastic Runge-Kutta algorithms. I. White noise. Phys Rev A 45:600–603

    Article  ADS  Google Scholar 

  27. Huopaniemi I, Luo K, Ala-Nissila T, Ying S-C (2006) Langevin dynamics simulations of polymer translocation through nanopores. J Chem Phys 125:124901

    Google Scholar 

  28. Jaeger HM, Nagel SR, Behringer RP (1996) The physics of granular materials. Phys Today 49:32

    Google Scholar 

  29. Jalalinejad A, Bassereh H, Salari V, Ala-Nissila T, Giacometti A (2018) Excitation energy transport with noise and disorder in a model of the selectivity filter of an ion channel. J Phys Condens Matter 30:415101

    Google Scholar 

  30. Jolliffe IT, Cadima J (2016) Principal component analysis: a review and recent developments. Philos Trans Royal Soc A 374:20150202

    Google Scholar 

  31. Jun Park P, Sung W (1999) Dynamics of a polymer surmounting a potential barrier: the Kramers problem for polymers. J Chem Phys 111:5259–5266

    Google Scholar 

  32. Kasianowicz JJ, Brandin E, Branton D, Deamer DW (1996) Characterization of individual polynucleotide molecules using a membrane channel. Proc Natl Acad Sci U.S.A 93:13770–13773

    Article  ADS  Google Scholar 

  33. Khinchin AI (1949) Mathematical foundations of statistical mechanics. Dover Publications

    Google Scholar 

  34. Klages R, Radons G, Sokolov IM (eds) (2008) Anomalous transport. Wiley & Sons

    Google Scholar 

  35. Klus S, Nüske F, Koltai P, Wu H, Kevrekidis I, Schütte C, Noé F (2018) Data-driven model reduction and transfer operator approximation. J Nonlinear Sci 28:985–1010

    Article  ADS  MathSciNet  Google Scholar 

  36. Lasanta A, Puglisi A (2015) An itinerant oscillator model with cage inertia for mesorheological granular experiments. J Chem Phys 143:064511

    Google Scholar 

  37. Lechenault F, Dauchot O, Biroli G, Bouchaud JP (2008) Critical scaling and hetherogeneous superdiffusion across the jamming/rigidity transition of a granular gas. Europhys Lett 83:46003

    Google Scholar 

  38. Lubensky DK, Nelson DR (1999) Driven polymer translocation through a narrow pore. Biophys J 77:1824–1838

    Article  Google Scholar 

  39. Mannella R, Palleschi V (1989) Fast and precise algorithm for computer simulation of stochastic differential equations. Phys Rev A 40:3381–3386

    Article  ADS  Google Scholar 

  40. Marty G, Dauchot O (2005) Subdiffusion and cage effect in a sheared granular material. Phys Rev Lett 94:015701

    Google Scholar 

  41. Melchionna S (2007) Design of quasisymplectic propagators for Langevin dynamics. J Chem Phys 127:044108

    Google Scholar 

  42. Meller A, Nivon L, Branton D (2001) Voltage-driven DNA translocations through a nanopore. Phys Rev Lett 86:3435

    Google Scholar 

  43. Miceli F, Baldovin M, Vulpiani A (2019) Statistical mechanics of systems with long-range interactions and negative absolute temperature. Phys Rev E 99:042152

    Google Scholar 

  44. Molgedey L, Schuster HG (1994) Separation of a mixture of independent signals using time delayed correlations. Phys Rev Lett 72:3634–3637

    Article  ADS  Google Scholar 

  45. Muthukumar M (2010) Theory of capture rate in polymer translocation. J Chem Phys 132:05B605

    Google Scholar 

  46. Noé F, Nuske F (2013) A variational approach to modeling slow processes in stochastic dynamical systems. Multiscale Model Simul 11:635–655

    Article  MathSciNet  Google Scholar 

  47. Onsager L, Machlup S (1953) Fluctuations and irreversible processes. Phys Rev 91:1505

    Google Scholar 

  48. Peinke J, Tabar MR, Wächter M (2019) The Fokker-Planck approach to complex spatiotemporal disordered systems. Annu Rev Condens Matter Phys 10:107–132

    Article  ADS  Google Scholar 

  49. Poincaré H, Magini R (1899) Les méthodes nouvelles de la mécanique céleste. Il Nuovo Cimento (1895-1900) 10:128–130

    Google Scholar 

  50. Polson JM, Hassanabad MF, McCaffrey A (2013) Simulation study of the polymer translocation free energy barrier. J Chem Phys 138:024906

    Google Scholar 

  51. Radjai F, Roux S (2002) Turbulentlike fluctuations in quasistatic flow of granular media. Phys Rev Lett 89:064302

    Google Scholar 

  52. Rahman A (1964) Correlations in the motion of Atoms in Liquid Argon. Phys Rev 136:A405

    Google Scholar 

  53. Reis PM, Ingale RA, Shattuck MD (2007) Caging dynamics in a granular fluid. Phys Rev Lett 98:188301

    Google Scholar 

  54. Rubin RJ (1960) Statistical dynamics of simple cubic lattices. Model for the study of Brownian motion. J Math Phys 1:309

    Google Scholar 

  55. Scalliet C, Gnoli A, Puglisi A, Vulpiani A (2015) Cages and anomalous diffusion in vibrated dense granular media. Phys Rev Lett 114:198001

    Google Scholar 

  56. Sebastian KL, Debnath A (2006) Polymer in a double well: dynamics of translocation of short chains over a barrier. J Phys Condensed Matter 18:S283–S296

    Article  ADS  Google Scholar 

  57. Sebastian K, Paul AK (2000) Kramers problem for a polymer in a double well. Phys Rev E 62:927–939

    Article  ADS  Google Scholar 

  58. Siegert S, Friedrich R (2001) Modeling of nonlinear Lévy processes by data analysis. Phys Rev E 64:041107

    Google Scholar 

  59. Tu JH, Rowley CW, Luchtenburg DM, Brunton SL, Kutz JN (2014) On dynamic mode decomposition: theory and applications. J Comput Nonlinear Dyn 1:391

    Google Scholar 

  60. Turner R (1960) Motion of a heavy particle in a one dimensional chain. Physica 26:269–273

    Article  ADS  MathSciNet  Google Scholar 

  61. Utter B, Behringer RP (2004) Self-diffusion in dense granular shear flows. Phys Rev E 69:031308

    Google Scholar 

  62. Verlet L (1967) Computer “Experiments” on classical fluids. I. thermodynamical properties of Lennard-Jones molecules. Phys Rev 159:98–103

    Article  ADS  Google Scholar 

  63. Wehmeyer C, NoÉ F (2018) Time-lagged autoencoders: deep learning of slow collective variables for molecular kinetics. J Chem Phys 148:241703

    Google Scholar 

  64. Wortel GH, Dijksman JA, van Hecke M (2014) Rheology of weakly vibrated granular media. Phys Rev E 89:012202

    Google Scholar 

  65. Zwanzig R (1961) Memory effects in irreversible thermodynamics. Phys Rev 124:983–992

    Article  ADS  Google Scholar 

  66. Zwanzig R (1973) Nonlinear generalized Langevin equations. J Stat Phys 9:215–220

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Sapienza University of Rome, Rome, Italy

    Marco Baldovin

Authors
  1. Marco Baldovin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Baldovin .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Baldovin, M. (2020). Computational and Technical Aspects. In: Statistical Mechanics of Hamiltonian Systems with Bounded Kinetic Terms. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-51170-8_6

Download citation

Publish with us

Policies and ethics

Search

Navigation