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Math Tools
Published in Thomas M. Nordlund, Peter M. Hoffmann, Quantitative Understanding of Biosystems, 2019
Thomas M. Nordlund, Peter M. Hoffmann
The previous section would seem to imply that distributions apply when the number of particles is large, perhaps approaching Avogadro’s number, 6.02 × 1023. The idea that a smooth distribution accurately describes the population of particles does require a large number of particles, or else a smaller number, even as small as 1, as long as adequate time is allowed by the experiment or biological process for the smaller number of particles to explore (pass through) the majority of the allowed and accessible states. These two extremes—ensembles of large numbers of particles whose properties are measured during a brief snapshot in time or one particle observed for an infinite length of time—constitute the ergodic hypothesis. A system is ergodic with respect to a measured property p if the distribution function describing that property is the same whether the ensemble average or the time average is measured: (
Bulk Stress in Particulate Dispersions and Composites
Published in Rajinder Pal, Rheology of Particulate Dispersions and Composites, 2006
A large number of configurations corresponding to the same macroscopic boundary conditions make up an ensemble. An ensemble average of any quantity is defined as the average taken over the various values occurring in these configurations. If q is some field variable (function of position x→ and time t) for some particular realization, ξ, of the process, then the ensemble average of q is () q¯(x→,t)=∫Mq(x→,t;ξ)dm
Molecular dynamics simulation
Published in Zhigang Li, Nanofluidics, 2018
“Ensemble” is a statistical mechanics term. An ensemble is a collection of systems, which are thermodynamically similar to a system of interest. The information of the systems in an ensemble is used to obtain the mean properties of the system under consideration. Depending on how a system interacts with its environment, the corresponding ensemble can be a microcanonical, canonical, or grand canonical ensemble.
An optical channel modeling of a single mode fiber
Published in Journal of Modern Optics, 2018
Neda Nabavi, Peng Liu, Trevor James Hall
The state of polarization of two light waves close in frequency slowly de-correlates with long distance transmissions (43). However, it has been proved that for two eigen modes the first order polarization dispersion is zero and they are called principal states of polarizations (9). These two orthogonal polarization states are frequency independent to first order. Considering each PSP to have a delay in the time domain, the difference between their group delays is called differential group delay. The data recorded can serve to help development of data analysis methods, as the theory indicates no information about the time behaviour of long term cross-correlations. This is because of the fact that theory uses ensemble averages, whereas measurement necessarily uses time averages. An assumption of ergodicity only implies that space averages would equal ensemble averages. But the length of the fibre cannot be varied in a practical measurement. Therefore, we shall consider modelling random processes by dynamical systems as considered in ergodic theory since the formulation and proof of ergodic theorems are more natural in the dynamical system context.
Tailoring geometric phases of two-dimensional functional materials under light: a brief review
Published in International Journal of Smart and Nano Materials, 2020
The Gibbs free energy, like internal energy in usual cases, is a thermodynamic averaged physical quantity. Under ergodic assumption, the ensemble average of is equivalent with time average. Thus, . In the following discussion, we will omit the notation, and time average is always taken. The time averaged Gibbs free energy is then