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Irreversibility of classical fluctuations studied in analogue electrical circuits

Nature volume 389, pages 463–466 (1997)Cite this article

Abstract

Fluctuations around some average or equilibrium state arise universally in physical systems. Large fluctuations — fluctuations that are much larger than average — occur only rarely but are responsible for many physical processes, such as nucleation in phase transitions, chemical reactions, mutations in DNA sequences, protein transport in cells and failure of electronic devices. They lie at the heart of many discussions1,2,3,4,5 of how the irreversible thermodynamic behaviour of bulk matter relates to the reversible (classical or quantum-mechanical) laws describing the constituent atoms and molecules. Large fluctuations can be described theoretically using hamiltonian6,7 and equivalent path-integral8,9,10,11,12 formulations, but these approaches remain largely untested experimentally, mainly because such fluctuations are rare and also because only recently was an appropriate statistical distribution function formulated11. It was shown recently, however, that experiments on fluctuations using analogue electronic circuits allow the phase-space trajectories of fluctuations in a dynamical system to be observed directly12. Here we show that this approach can be used to identify a fundamental distinction between two types of random motion: fluctuational motion, which takes the system away from a stable state, and relaxational motion back towards this state. We suggest that macroscopic irreversibility is related to temporal asymmetry of these two types of motion, which in turn implies a lack of detailed balance and corresponds to non-differentiability of the generalized nonequilibrium potential in which the motion takes place.

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Figure 1: a–d, Fluctuational behaviour measured and calculated for an electronic model system in equilibrium: a double-well Duffing oscillator with K(x) = x − x3, for D = 0.014.
Figure 2: a–d, Fluctuational behaviour measured and calculated for an electronic model system in equilibrium: a double-well Duffing oscillator with K(x) = x − x3, for D = 0.014.
Figure 3: a–d, Fluctuational behaviour measured and calculated for an electronic model system in equilibrium: a double-well Duffing oscillator with K(x) = x − x3, for D = 0.014.
Figure 4: a–d, Fluctuational behaviour measured and calculated for an electronic model system in equilibrium: a double-well Duffing oscillator with K(x) = x − x3, for D = 0.014.
Figure 5: a–c, Fluctuational behaviour measured and calculated for an electronic model of a nonequilibrium system with explicit time dependence: an overdamped double-well Duffing oscillator with K(x,t) = x − x3 + Acos(ωt) and A = 0.264, D = 0.012.
Figure 6: a–c, Fluctuational behaviour measured and calculated for an electronic model of a nonequilibrium system with explicit time dependence: an overdamped double-well Duffing oscillator with K(x,t) = x − x3 + Acos(ωt) and A = 0.264, D = 0.012.
Figure 7: a–c, Fluctuational behaviour measured and calculated for an electronic model of a nonequilibrium system with explicit time dependence: an overdamped double-well Duffing oscillator with K(x,t) = x − x3 + Acos(ωt) and A = 0.264, D = 0.012.
Figure 8: Fluctuational behaviour measured and calculated for an electronic model of a nonequilibrium system with a stationary nongradient field: K(x,y)=(x−x3−axy2,−(1+x2)y); a=10; D=0.014.
Figure 9: Fluctuational behaviour measured and calculated for an electronic model of a nonequilibrium system with a stationary nongradient field: K(x,y)=(x−x3−axy2,−(1+x2)y); a=10; D=0.014.
Figure 10: Fluctuational behaviour measured and calculated for an electronic model of a nonequilibrium system with a stationary nongradient field: K(x,y)=(x−x3−axy2,−(1+x2)y); a=10; D=0.014.

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Acknowledgements

We thank M. I. Dykman for help and encouragement; we also thank him, R. S. Maier and D. L. Stein for comments on an earlier version of the manuscript; and we acknowledge correspondence with N. G. van Kampen. This work was supported by the Engineering and Physical Sciences Research Council (UK), the Royal Society, and the Russian Foundation for Basic Research.

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Authors and Affiliations

  1. School of Physics and Chemistry, Lancaster University, Lancaster, LA1 4YB, UK

    D. G. Luchinsky & P. V. E. McClintock

  2. Russian Research Institute for Metrological Service, Ozernaya 46, Moscow, 119361, Russia

    D. G. Luchinsky

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Correspondence to P. V. E. McClintock.

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Luchinsky, D., McClintock, P. Irreversibility of classical fluctuations studied in analogue electrical circuits. Nature 389, 463–466 (1997). https://doi.org/10.1038/38963

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