Temperature relaxation in strongly-coupled binary ionic mixtures

New facilities such as the National Ignition Facility and the Linac Coherent Light Source have pushed the frontiers of high energy-density matter. These facilities offer unprecedented opportunities for exploring extreme states of matter, ranging from cryogenic solid-state systems to hot, dense plasmas, with applications to inertial-confinement fusion and astrophysics. However, significant gaps in our understanding of material properties in these rapidly evolving systems still persist. In particular, non-equilibrium transport properties of strongly-coupled Coulomb systems remain an open question. Here, we study ion-ion temperature relaxation in a binary mixture, exploiting a recently-developed dual-species ultracold neutral plasma. We compare measured relaxation rates with atomistic simulations and a range of popular theories. Our work validates the assumptions and capabilities of the simulations and invalidates theoretical models in this regime. This work illustrates an approach for precision determinations of detailed material properties in Coulomb mixtures across a wide range of conditions.

In plasma physics the temperature is often defined pragmatically as This arises from a discretization of the distribution function where f (v, t) is the microscopic form of the velocity distribution In homogeneous plasmas there is no drift, u = v = 0 and Eq. (1) is recovered. The temperature could be defined alternatively by fitting the binned one-dimensional velocity distribution to a Maxwellian, with σ as the fit parameter. The temperature would then be k B T = mσ 2 . As the system approaches equilibrium, these two definitions of temperature will agree. The extent to which they do not agree is a measure of the non-equilibrium nature of the system. Quantifying the departure of the temperature defined using Eq. (1) from the equilibrium value can be done using a Hermite polynomial expansion (1,2). For a normalized one-dimensional velocity distribution where H n (v/σ) are the probabilistic Hermite polynomials of order n and the Maxwellian distribution acts as a weight function. The Hermite coefficients are found using orthogonality relations.
where the time dependence has been suppressed for brevity. Critical to Hermite analysis is a correct value for the width of the Maxwellian weight function σ. When the distribution function f (v, t) is itself a Maxwellian distribution with rms width equal to σ, the Hermite coefficients a n will all vanish except for a 0 = 1. However, if f (v, t) is Maxwellian with a width different from σ, the expansion in Eq. 5 systematically gives non-zero values for the a n coefficients.
When the value of σ is not known beforehand, it can be estimated using the second moment of the velocity distribution, While this is a logical choice, it complicates the Hermite analysis because σ is a parameter in the Hermite polynomials. In particular, the second moment can be rewritten as, The width of the weight function σ and the Hermite coefficient a 2 are coupled together. Just as v 2 /σ 2 = 1 + a 2 , higher moments of the velocity distribution are also linear combinations of the Hermite coefficients, e.g. v 4 /σ 2 = 3a 0 + 6a 2 + a 4 . Without an unambiguous determination of σ, the Hermite coefficients cannot be determined from the velocity moment calculations alone.
It might be assumed that the values of σ and a n could be extracted directly from a fit of f (v, t) to Eq. (5). This is reliable only when the noise distribution in each velocity bin of f (v, t) is Gaussian with zero mean. However, the MD distributions are necessarily non-negative. Towards the wings of the distribution, where there are relatively few particles per bin, the noise distribution in each bin becomes binomial and has a non-zero mean. This feature causes stochastic changes in the determination of σ and a 2 because of the coupling relationship in Eq. (9). It also artificially biases the determination of a n with n ≥ 4. It might be possible to artificially augment the MD noise distribution in each velocity bin in an attempt to force the condition of zero mean. However, the reliability of this approach is not known. This remains an area of future research.
As discussed in the main text, ratios of the velocity moments give insight into non-Maxwellian departures of the distribution. In the case of a perfect Maxwellian distribution moments higher than n = 2 satisfy the relation v n = 0 if n odd, ( v 2 ) n/2 (n − 1)!! if n even. (10) Therefore the moment ratios v 4 /3 v 2 2 , v 6 /15 v 2 3 , and v 8 /105 v 2 4 are all equal to unity. Calculations of moment ratios and Hermite coefficients are influenced by the small numbers of particles in the wings of the MD distribution at large velocities. The values of the coefficients and moment ratios therefore depend on how far out into the wings the analysis persists. The moment ratios included in the manuscript span the distribution out to ±7σ, where σ is estimated using a Gaussian fit to the MD distribution.