We experimentally demonstrate the fluctuation theorem, which predicts appreciable and measurable violations of the second law of thermodynamics for small systems over short time scales, by following the trajectory of a colloidal particle captured in an optical trap that is translated relative to surrounding water molecules. From each particle trajectory, we calculate the entropy production/consumption over the duration of the trajectory and determine the fraction of second law-defying trajectories. Our results show entropy consumption can occur over colloidal length and time scales.

The free energy landscape and the folding mechanism of the C-terminal ␤-hairpin of protein G is studied by extensive replica exchange molecular dynamics simulations (40 replicas and 340 ns total simulation time), using the GROMOS96 force field and the SPC explicit water solvent. The study reveals that the system preferentially adopts a ␤-hairpin structure at biologically important temperatures, and that the helix content is low at all temperatures studied. Representing the free energy landscape as a function of several types of reaction coordinates, four local minima corresponding to the folded, partially folded, molten globule, and unfolded states are identified. The findings suggest that the folding of the ␤-hairpin occurs as the sequence: collapse of hydrophobic core 3 formation of H-bond 3 formation of the turn. Identifying the folded and molten globule states as the main conformations, the free energy landscape of the ␤-hairpin is consistent with a two-state behavior with a broad transition state. The temperature dependence of the folding-unfolding transition is investigated in some detail. The enthalpy and entropy jumps at the folding transition temperature are found to be about three times lower than the experimental estimates, indicating that the folding-unfolding transition in silico is less cooperative than its in vitro counterpart. Proteins 2005;61:795-808.

For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields. DOI: 10.1103/PhysRevE.63.051105 PACS number͑s͒: 05.20.Ϫy, 47.10.ϩg The fluctuation theorem ͓1-3͔ ͑FT͒ gives a general formula for the logarithm of the probability ratio that in a thermostated dissipative system, the time-averaged entropy production ⌺ t takes a value A to minus the value, ϪA,From this equation it is obvious that as the averaging time or system size increases, it becomes exponentially likely that the entropy production will be positive. The fluctuation theorem is important for at least three reasons: first, it gives an expression for the probability that in a finite system observed for a finite time, the Second Law will be violated; secondly, it gives one of the very few exact fluctuation relations that are known for nonequilibrium steady states, even far from equilibrium; thirdly, it can be derived using some of the standard results of the mathematical theory of dynamical systems theory ͓3͔. The theorem was initially proposed ͓1͔ for nonequilibrium steady states that are thermostated in such a way that the total energy of the system is constant. Subsequently, it was shown by Gallavotti and Cohen ͓3͔ that the theorem could be proved for sufficiently chaotic, isoenergetic nonequilibrium systems using the Sinai-Ruelle-Bowen measure ͓3͔. For transient trajectory segments that start at tϭ0 from an ensemble of initial phase-space vectors ⌫͑0͒, and evolve in time under the influence of a reversible deterministic thermostat and an applied dissipative field towards a unique nonequilibrium steady state, a transient fluctuation theorem can be derived using the Liouville measure ͓2,4͔. It has also been shown that the theorem is valid for a wide class of stochastic nonequilibrium systems ͓5͔.It has recently be shown ͓4͔ that if initial phases are sampled from a known N-particle phase-space distribution function, f (⌫,0), and if we define a dissipation function, ⍀͑⌫͒, bywhere ‫⌫•⌫ץ/ץ‪⌳(⌫)ϵ‬‬ is the phase-space compression factor, then one can derive a transient fluctuation theorem,Thermostats lead to nonzero expressions for the phase-space compression factor. Equation ͑2͒ is consistent with all known deterministic transient fluctuation theorems covering a wide variety of initial ensemble types and thermostating mechanisms ͓4͔.This relationship ͑3͒ has been tested using computer simulations for a range of nonequilibrium steady-state systems in which the phase-space contraction is nonzero ͓2,4,5͑b͒,6͔. It predicts t...

The Fluctuation Theorem (FT) is a generalisation of the Second Law of Thermodynamics that applies to small systems observed for short times. For thermostatted systems it gives the probability ratio that entropy will be consumed rather than produced. In the present paper, we propose a version of the FT, that applies to thermostatted dissipative systems which respond to time dependent dissipative fields. In testing the time dependent Fluctuation Theorem we provide for the first time, convincing evidence that sets of trajectories with conjugate values for the time integrated entropy production, (±A±dA), are indeed (for time reversible dynamical systems such as studied here), time reversal images of one another. This observation verifies the deep connection between time reversal symmetry, the Fluctuation Theorem and the Second Law of Thermodynamics.

Linear irreversible thermodynamics asserts that the instantaneous local spontaneous entropy production is always nonnegative. However, for a viscoelastic fluid this is not always the case. Given the fundamental status of the Second Law, this presents a problem. We provide a new rigorous derivation of the Second Law, which is valid for the appropriately time averaged entropy production allowing the instantaneous entropy production to be negative for short intervals of time. We show that time averages (rather than instantaneous values) of the entropy production are nonnegative. We illustrate this using molecular dynamics simulations of oscillatory shear.