Chapter 17
Tensor Dynamics
Abstract This chapter presents examples for dynamical phenomena involving ten-
sors. Firstly, linear tensor equations are considered which provide the basis for the
computation of time-correlation functions and of spectral functions describing the
frequency dependence, e.g. of scattered radiation. Secondly, nonlinear relaxation
phenomena involving the second rank alignment tensor are treated. Basis tensors are
introduced which lead to coupled non-linear equations for the relevant components
of the tensor. The stability of stationary solutions is analyzed. Thirdly, the effect of an
imposed shear flow on the alignment tensor is considered. Depending on the model
parameters, stationary as well as periodic and chaotic solutions are obtained. Similar
features are found for a nonlinear Maxwell model governing the shear stress tensor.
17.1 Time-Correlation Functions and Spectral Functions
The dynamics of small fluctuations about an equilibrium state, as well as of small
macroscopic deviations from equilibrium are described by time-correlation func-
tions. Spectral functions are obtained by a time-Fourier transformation.
17.1.1 Definitions.
Letψi =ψi(t)withi = 1 , 2 ,...be functions which depend on the timetvia
their dependence on dynamic variables like the position, the linear momentum or the
internal angular momentum of a particle. Appropriately defined averages〈...〉 0 of
these quantities are assumed to vanish, viz.〈ψi〉 0 =0. In general, theψifluctuate
about their average values, their squares averaged are non-zero:〈ψi^2 〉 0 >0. The
average
Cij(t)=〈
ψi(t 0 +t)ψj(t 0 )〉
0 , i=^1 ,^2 ,..., j=^1 ,^2 ,..., (17.1)© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_17
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