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Fluctuation Theorems, Brownian Motors and Thermodynamics of Small Systems

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

The thermodynamics of small systems describes energy exchange processes between a system and its environment in the low energy range ofa few \( { k_{\mathrm{B}}T } \) whereBrownian fluctuations are dominant [1]. The main goal of this discipline is to identify the buildingblocks of a general theory describing energy fluctuations in non‐equilibrium processes occurring in systems ranging from condensed matterphysics to biophysics.

Thermodynamics, a scientific discipline inherited from the 18th century, is facing new challenges in the description of non‐equilibriumsmall (sometimes also called mesoscopic) systems. Thermodynamics is a discipline built in order to explain and interpret energetic processesoccurring in macroscopic systems made out of a large number of molecules on the order of the Avogadro number. The subsequent development ofstatistical mechanics has provided a solid probabilistic basis to thermodynamics...

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Abbreviations

Trajectory or path:

A crucial concept in the statistical description of small system is that of a trajectory or path. A path is the time sequence of configurations followed by the system as it is driven to a non‐equilibrium state by the action of an external perturbation.

Control parameter:

External perturbations are usually described in terms of the control parameter λ. These are a set of external parameters (e. g. an electric field, magnetic field, optical force, …) that can be experimentally controlled and do not fluctuate. Experimentally, control parameters are produced by macroscopic systems that are used to manipulate the small system under study and which are insensitive to thermal fluctuations (but that produce other sorts of uncontrolled instrumental noises and drift effects).

Single molecule experiments (SME):

Recent technological developments have provided the tools to design and build scientific instruments of high enough sensitivity and precision to manipulate and visualize individual molecules and measure microscopic forces. Using SME it is possible to manipulate molecules one at a time and measure distributions describing molecular properties, characterize the kinetics of bio‐molecular reactions, and detect molecular intermediates. SME provide the additional information about thermodynamics and kinetics of bio‐molecular processes. This complements information obtained in traditional bulk assays. In SME it is also possible to measure small energies and detect large Brownian deviations in bio‐molecular reactions, thereby offering new methods and systems to scrutinize the basic foundations of statistical mechanics. Common single molecule experimental techniques are: atomic‐force microscopy, laser optical tweezers, magnetic tweezers and single‐molecule fluorescence.

Free energy:

The natural or spontaneous evolution of any thermodynamic process is determined by the free energy. The free energy in thermodynamics is the equivalent of the mechanical energy in classical mechanics. Spontaneous transformations take place by a decrease of the free energy in the system. In addition, mechanical work must be exerted by an external agent upon the system to increase its free energy. For reversible processes the amount of work is equal to the free energy change. However, in general, processes are irreversible and the work must be always larger than the free energy difference (a statement of the second law of thermodynamics). Free energies in small systems are typically expressed in either work (pN·nm) or energy units (kJ/mol, kcal/mol or \( { k_{\mathrm{B}}T } \) where \( { k_{\mathrm{B}} } \) is the Boltzmann constant and T is a reference temperature – usually 298 K or 25 degrees Celsius). The conversion factors are \( (T=298\,\mathrm{K})\colon 1\,k_{\mathrm{B}}T= 4.11\,\mathrm{pN}\cdot\mathrm{nm}=4.11 10^{-21}\,\mathrm{J} \), \( 1\,k_{\mathrm{B}}T=0.6\,\mathrm{kcal}/\mathrm{mol}=2.4\,\mathrm{kJ}/\mathrm{mol} \).

ATP:

Acronym for adenosine triphosphate, the molecule that carries the energy necessary to sustain life processes. ATP is made of one adenosine base weakly bonded to three phosphate groups. Upon conversion (by hydrolysis) to ADP (adenosine diphosphate) and inorganic phosphate or AMP (adenosine monophosphate) and pirophosphate (P-P), ATP delivers a considerable amount of free energy (in the range 8–12 kcal/mol, depending on buffer conditions). By coupling to other reactions, ATP hydrolysis supplies the energy necessary to carry out unfavorable transformations.

RNA:

RNA (ribonucleic acid) is a very important player in molecular biology that shows biological functions in between those attributed to DNA and proteins. For the biophysicist and the statistical physicist RNA is also a fascinating molecule. Primarily found in nature in single stranded form, RNA folds into a three dimensional structure mainly stabilized by stacking interactions and hydrogen bonds between complementary bases (A-U,G-C). Full complementarity between different RNA segments is often impossible so, at difference with DNA, RNA structure includes also mismatches between bases as well other structural defects (bulges, loops, junctions, …). In addition to Watson–Crick base pairing, RNA forms a compact structure through specific interactions mediated by magnesium ions that bring together distal RNA segments.

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Authors and Affiliations

  1. Department de Fisica Fonamental, Faculty of Physics, Universitat de Barcelona, Barcelona, Spain

    Felix Ritort

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  1. Felix Ritort
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Ritort, F. (2009). Fluctuation Theorems, Brownian Motors and Thermodynamics of Small Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_213

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