Heat fluctuations are studied in a dissipative system with both deterministic and stochastic components for a simple model: a Brownian particle dragged through water by a moving potential. An extension of the stationary state fluctuation theorem is derived. For infinite time, this reduces to the conventional fluctuation theorem only for small fluctuations; for large fluctuations, it gives a much larger ratio of the probabilities of the particle to absorb rather than supply heat. This persists for finite times and should be observable in experiments similar to a recent one carried out by Wang et al.

Recently Wang et al. carried out a laboratory experiment, where a Brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center. This experiment confirmed a theoretically predicted work related integrated transient fluctuation theorem (ITFT), which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state. The corresponding integrated stationary state fluctuation theorem (ISSFT) was not observed. Using an overdamped Langevin equation and an arbitrary motion for the center of the harmonic force, all quantities of interest for these theorems and the corresponding nonintegrated ones (TFT and SSFT, respectively) are theoretically explicitly obtained in this paper. While the TFT and the ITFT are satisfied for all times, the SSFT and the ISSFT only hold asymptotically in time. Suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the ISSFT could be observed, are given. In addition, a nontrivial long-time relation between the ITFT and the ISSFT was discovered, which could be observed experimentally, especially in the case of a resonant circular motion of the center of the harmonic force.

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Heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution of these fluctuations for all tau. By a saddle-point method we obtain analytical results for the inverse Fourier transform, which, for not too small tau, agree very well with numerical results from a sampling method as well as from the fast Fourier transform algorithm. Due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation theorem is, for infinite and for finite tau, replaced by an extended fluctuation theorem that differs noticeably and measurably from it. In particular, for large fluctuations, the ratio of the probability for absorption of heat (by the particle from the fluid) to the probability to supply heat (by the particle to the fluid) is much larger here than in the conventional fluctuation theorem.

Using recent fluctuation theorems from nonequilibrium statistical mechanics, we extend the theory for voltage fluctuations in electric circuits to power and heat fluctuations. They could be of particular relevance for the functioning of small circuits. This is done for a parallel resistor and capacitor with a constant current source for which we use the analogy with a Brownian particle dragged through a fluid by a moving harmonic potential, where circuit-specific analogues are needed on top of the Brownian-Nyquist analogy. The results may also hold for other circuits as another example shows. Nanotechnology is quickly getting within reach, but the physics at these scales could be different from that at the macroscopic scale. In particular, large fluctuations will occur, with mostly unknown consequences. In this letter, we will investigate properties of electric circuits concerning the fluctuations of power and heat within the context of the so-called Fluctuation Theorems (FTs). These theorems were originally found in the context of nonequilibrium dynamical systems theory. Surprisingly, these can be applied also to electric circuits, as we will show, and thus give further insight into their behavior.Let us first give a brief introduction to the FTs. First found in dynamical systems[1, 2] and later extended to stochastic systems[3], these conventional FTs give a relation between the probabilities to observe a positive value of the (time averaged) "entropy production rate" and a negative one. This relation is of the form P (σ)/P (−σ) = exp [στ ], where σ and −σ are equal but opposite values for the entropy production rate, P (σ) and P (−σ) give their probabilities and τ is the length of the interval over which σ is measured. In these systems, the above mentioned FT is derived for a mathematical quantity σ, which has a form similar to that of the entropy production rate in Irreversible Thermodynamics.Apart from an early experiment in a turbulent flow [4], for quite some time, the investigations of the FTs were restricted to theoretical approaches and simulations. In 2002, Wang et al. performed an experiment on a micron-sized Brownian particle dragged through water by a moving optical tweezer. In this experiment, a Transient Fluctuation Theorem (TFT) was demonstrated for fluctuations of the total external work done on the system in the transient state of the system, i.e., considering a time interval of duration τ which starts immediately after the tweezer has been set in motion [5]. In contrast, a Stationary State Fluctuation Theorem (SSFT), which was not measured, would concern fluctuations in the stationary state, i.e., in intervals of duration τ starting at a time long after the tweezer has been set in motion. While the work fluctuations satisfy the conventional TFT and SSFT [6,7], the heat fluctuations satisfy different, extended FTs due to the interplay of the stochastic motion of the fluid with the deterministic harmonic potential induced by the optical tweezer [8,9]. Given the possible problems with identifyin...

The evolution of a mixed quantum-classical system is expressed in the mapping formalism where discrete quantum states are mapped onto oscillator states, resulting in a phase space description of the quantum degrees of freedom. By defining projection operators onto the mapping states corresponding to the physical quantum states, it is shown that the mapping quantum-classical Liouville operator commutes with the projection operator so that the dynamics is confined to the physical space. It is also shown that a trajectory-based solution of this equation can be constructed that requires the simulation of an ensemble of entangled trajectories. An approximation to this evolution equation which retains only the Poisson bracket contribution to the evolution operator does admit a solution in an ensemble of independent trajectories but it is shown that this operator does not commute with the projection operators and the dynamics may take the system outside the physical space. The dynamical instabilities, utility and domain of validity of this approximate dynamics are discussed. The effects are illustrated by simulations on several quantum systems.

The largest Lyapunov exponent l 1 for a dilute gas with short range interactions in equilibrium is studied by a mapping to a clock model, in which every particle carries a watch, with a discrete time that is advanced at collisions. This model has a propagating front solution with a speed that determines l 1 , for which we find a density dependence as predicted by Krylov, but with a larger prefactor. Simulations for the clock model and for hard sphere and hard disk systems confirm these results and are in excellent mutual agreement. They show a slow convergence of l 1 with increasing particle number, in good agreement with a prediction by Brunet and Derrida. [S0031-9007 (98) Recently, there has been great interest in the relationship between statistical mechanics and the theory of dynamical systems [1][2][3]. Calculating dynamical properties such as Lyapunov exponents for statistical mechanical systems usually requires numerical simulations. For the Lorentz gas, however, Dorfman, Van Beijeren, and others [3,4] have obtained analytical expressions for the Lyapunov spectrum and Kolmogorov-Sinai entropy at low densities, both in equilibrium and for the field-driven case.In this paper we present an analytic calculation of the largest Lyapunov exponent in the low density limit for a gas at equilibrium consisting of particles with short range interactions. Our method is based on arguments from kinetic theory and similar in spirit to the method of Refs. [3,4]. We compare our results to those from computer simulations on hard disk and hard sphere systems and pay special attention to the dependence of the largest Lyapunov exponent on the total number of particles.We consider a gas consisting of N atoms of diameter s, defined as the (strictly finite) range of interaction, and mass m in d dimensions, in a volume V . The reduced densityñ is defined as Ns d ͞V and will serve as a small parameter. To calculate the largest Lyapunov exponent we follow two nearby trajectories in phase space. For the first one, the reference trajectory, the positions and velocities of the particles are denoted by ͑ r i , y i ͒. In the second trajectory they are denoted by ͑ r i 1 d r i

Niagara is currently the fastest supercomputer accessible to academics in Canada. It was deployed at the beginning of 2018 and has been serving the research community ever since. This homogeneous 60,000-core cluster, owned by the University of Toronto and operated by SciNet, was intended to enable large parallel jobs and has a measured performance of 3.02 petaflops, debuting at #53 in the June 2018 TOP500 list. It was designed to optimize throughput of a range of scientific codes running at scale, energy efficiency, and network and storage performance and capacity. It replaced two systems that SciNet operated for over 8 years, the Tightly Coupled System (TCS) and the General Purpose Cluster (GPC) [13]. In this paper we describe the transition process from these two systems, the procurement and deployment processes, as well as the unique features that make Niagara a one-of-a-kind machine in Canada.