Equilibrium properties in statistical physics are obtained by computing averages with respect to Boltzmann-Gibbs measures, sampled in practice using ergodic dynamics such as the Langevin dynamics. Some quantities however cannot be computed by simply sampling the Boltzmann-Gibbs measure, in particular transport coefficients, which relate the current of some physical quantity of interest to the forcing needed to induce it. For instance, a temperature difference induces an energy current, the proportionality factor between these two quantities being the thermal conductivity. From an abstract point of view, transport coefficients can also be considered as some form of sensitivity analysis with respect to an added forcing to the baseline dynamics. There are various numerical techniques to estimate transport coefficients, which all suffer from large errors, in particular large statistical errors. This contribution reviews the most popular methods, namely the Green-Kubo approach where the transport coefficient is expressed as some time-integrated correlation function, and the approach based on longtime averages of the stochastic dynamics perturbed by an external driving (so-called nonequilibrium molecular dynamics). In each case, the various sources of errors are made precise, in particular the bias related to the time discretization of the underlying continuous dynamics, and the variance of the associated Monte Carlo estimators. Some recent alternative techniques to estimate transport coefficients are also discussed.
翻译:统计物理中的平衡特性是通过计算Boltzmann-Gibbs测量值的平均值获得的,在实际中采用Langevin动态等电子动态进行抽样,但有些数量无法通过仅仅抽样调查Boltzmann-Gibbs测量值,特别是运输系数来计算,这些系数将某些实际数量的当前利益与诱发该系数所需的强制力联系起来。例如,温度差异导致一种能量流,这两个数量之间的比例系数是热导导力。从抽象的角度来看,运输系数也可以被视为对基准动态增加的强迫力进行某种形式的敏感度分析。有各种估算运输系数的数字技术,所有这些都存在很大的错误,特别是巨大的统计错误。这一贡献审查了最受欢迎的方法,即格林-库博方法,其中运输系数表现为某种时间整合的关联功能,以及基于长期平均温度变化因外部驱动而受扰动(即所谓的无均匀分子动态)影响的方法。在每种情况下,各种变化系数的预测值都是与连续变化变化技术有关的原因。