Thermodynamics of a Colloidal Particle in a Time-Dependent Nonharmonic Potential

Abstract

We study the motion of an overdamped colloidal particle in a time-dependent nonharmonic potential. We demonstrate the first lawlike balance between applied work, exchanged heat, and internal energy on the level of a single trajectory. The observed distribution of applied work is distinctly non-Gaussian in good agreement with numerical calculations. Both the Jarzynski relation and a detailed fluctuation theorem are verified with good accuracy. DOI: 10.1103/PhysRevLett.96.070603 PACS numbers: 05.40.ÿa, 05.70.ÿa, 82.70.Dd Since more than a century, the first law relating the work applied to a system with both the exchanged heat and an increase in internal energy is one of the cornerstones of macroscopic physics. Its consistent formulation for a mesoscopic system like a driven colloidal particle, however, was suggested only about a decade ago [1]. Since on these scales thermal fluctuations are relevant, probability distributions for work, heat, and internal energy replace the sharp values of their macroscopic counterparts. Various theoretical relations like the fluctuation theorem [2,3], the Jarzynski relation [4], and the Hatano-Sasa relation [5] involving these distributions in different settings extend the second law to the mesoscopic realm at least as long as the notion of a constant temperature of the ambient heat bath remains meaningful [for a review, see [6]]. Such theorems have been tested experimentally using both biomolecules manipulated mechanically [7,8] as well as colloidal particles in time dependent laser traps [9][10][11]. Common to all colloidal experiments, so far, is that these laser traps generate a harmonic potential albeit with a timedependent center or ''spring constant.'' Consequently, often the interesting distributions are Gaussian even though for certain quantities non-Gaussian distributions can occur [10,12].In this Letter, we study the thermodynamics of single colloidal trajectories in a time-dependent nonharmonic potential which, generically, gives rise to non-Gaussian distributions. Only for very short or very long trajectories, one expects Gaussian distributions even in this nonharmonic case [13]. In particular, we identify applied work, exchanged heat, and change in internal energy along a single trajectory and thus test the consistency of these notions on this level, or, put differently, illustrate the validity of the first law. We measure the distribution of work in the non-Gaussian regime and compare it to theoretical prediction. Such a comparison does not involve a single fit parameter since all quantities are measured independently, which is another advantage of colloidal systems. Finally, we test the Jarzynski relation which expresses the free energy difference between two equilibrium states in terms of the nonequilibrium work spent in the transition between the two states. Such an illustration of the Jarzynski relation in the non-Gaussian regime comes timely given ongoing theoretical criticism of its validity [14,15]. In our study, particle trajectories were determined usi...

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