Single-time relaxation process

As we have seen, the time dependence of a macroscopic relaxation process always reflects the underlying microscopic dynamics. We may now proceed and look for kinetical equations which correctly describe the time dependence of the observed retarded responses. 

There is an obvious choice for the simple case when only a single characteristic time is included. It goes back to Debye, who proposed it in a famous work on the dielectric properties of polar liquids, based on a statistical mechanical theory. We formulate the equation for the above mentioned simple mechanical relaxation process, associated with transitions between two conformational states only, and consider a creep experiment under shear stress. 

It represents a linear differential equation of the first order, implying the assumption that, for a system in non-equilibrium, relaxation takes place with a rate which increases linearly with the distance from the equilibrium state. This is not a specific kind of expression devised to deal exclusively with our problem. Equivalent equations are broadly used in thermodynamics to describe the kinetics of all sorts of irreversible processes. Importantly, the equation includes one time constant only, the relaxation time r. 

The solution of the relaxation equation for the creep experiment, i.e. a step-like application of a stress at zero time, can be written down directly. It is given by 

The relaxation equation (5.60) is not restricted in use to step-like changes in the external conditions, but also holds, if the equilibrium value AJdzx is not a constant and changes with time. In particular, it can be employed to treat dynamic-mechanical experiments. As is clear, applying an oscillatory shear stress 

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The dynamic compliance of the single-time relaxation process, in the literature also addressed as Debye-process , thus has a simple form, being a function of the product cur and A J only. Separation into the real and the imaginary part yields... 

A simple check, if a measured dynamic compliance or a dielectric function agrees with a Debye-process, is provided by the Cole-Cole plot . Let us illustrate it with a dielectric single-time relaxation process. If we choose for the dipolar polarization an expression analogous to Eq. (5.66) and take also into account the instantaneous electronic polarization with a dielectric constant u, the dielectric function e uj) shows the form... 

Having established the properties of the single-time relaxation process, we have now also a means to represent a more complex behavior. This can be accomplished by applying the superposition principle, which must always hold in systems controlled by linear equations. Considering shear properties again, we write for a dynamic compliance J (uj) with general shape a sum of Debye-processes with relaxation times ti and relaxation strengths AJi... 

Rather than representing the viscoelastic properties of a given sample in the form of Eq. (5.73), i.e. by a superposition of Debye-processes which are specified by AJ/ and r, one can perform an analogous procedure based on single-time relaxation processes specified by AG/ and f/. We then write in the integral form... 

Neither the a-process nor the normal mode equal a single-time relaxation process. A good representation of data is often achieved by use of the empirical Havriliak-Nagami equation which has the form... 

It follows that there are two kinds of processes required for an arbitrary initial state to relax to an equilibrium state the diagonal elements must redistribute to a Boltzmaim distribution and the off-diagonal elements must decay to zero. The first of these processes is called population decay in two-level systems this time scale is called Ty The second of these processes is called dephasmg, or coherence decay in two-level systems there is a single time scale for this process called T. There is a well-known relationship in two level systems, valid for weak system-bath coupling, that... 

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