Critical dynamics of an isothermal compressible nonideal fluid

A pure fluid at its critical point shows a dramatic slow-down in its dynamics, due to a divergence of the order-parameter susceptibility and the coefficient of heat transport. Under isothermal conditions, however, sound waves provide the only possible relaxation mechanism for order-parameter fluctuations. Here we study the critical dynamics of an isothermal, compressible nonideal fluid via scaling arguments and computer simulations of the corresponding fluctuating hydrodynamics equations. We show that, below a critical dimension of 4, the order-parameter dynamics of an isothermal fluid effectively reduces to “model A,” characterized by overdamped sound waves and a divergent bulk viscosity. In contrast, the shear viscosity remains finite above two dimensions. Possible applications of the model are discussed.

Figure 1

Typical shape of the dynamic structure factor $C(k,\omega )$ of an isothermal fluid in the weak-damping (a) and strong-damping (b) regime. The latter case is realized at the critical point (see text). For comparison, (c) shows the structure factor of an ordinary, energy-conserving fluid close to the critical point.

Figure 2

Fundamental vertices of the model arising from Eqs. (20, 21, 22). Here, $\mathbf{k}$ is an external wave vector, while $\mathbf{q}$ and $\sigma$ denote an internal wave vector and frequency. A solid circle represents a coupling constant and an integration over internal wave vectors and frequencies respecting space- and time-translational invariance.

Figure 3

Leading frequency-dependent diagrams contributing to the self-energy of the order parameter, $\Sigma (\mathbf{k},\omega )$. A solid (wavy) line with an arrow represents $G_0$ ($G_{t,0}$) and a solid (wavy) line with an open circle $C_0$ ($C_{t,0}$). Dashed lines indicate amputated external “legs” for clarity. Note that only certain combinations of couplings are admissible in the diagrams of ${\Sigma }_j$ (cf. Fig. 2).

Figure 4

[(a) and (b)] Dynamic structure factor at the critical point: (a) Raw data for the lowest wave numbers $k$ (increasing from top to bottom). The time is given in lattice units and the lines are drawn as a guide to the eye. (b) Test of the dynamic scaling form of $C(k,t)$. The value of the dynamic critical index $z$ is determined by varying $z$ until all points fall onto a master curve. The inset shows that the collapse is incomplete for $z$ significantly differing from $2.25$. (c) Relaxation rate $\Gamma$ at the critical point obtained from exponential fits to the dynamic structure factor for different wave numbers. The solid line is $\propto k^{2.2}$.

Figure 5

(a) Dependence of the relaxation rate on the reduced temperature $\theta =(r_c-r)/r_c$ ($r$ is the Landau parameter and $r_c$ its value at the critical point) for different wave numbers. The expected power-law decrease of $\Gamma$ (dashed line) is rounded off when approaching the critical temperature due to the finite system size. (b) Wave-number dependence of the relaxation rate for different reduced temperatures approaching the critical point from above. Data points in the transition region between the overdamped and propagating acoustic regime are omitted. The power law $\sim k^{2-\eta }\sim k^{1.75}$ corresponds to the van Hove prediction for $\Gamma$, where a wave-number-independent ${\nu }_{lR}$ is assumed. (c) Effective longitudinal viscosity ${\nu }_{lR}$ (normalized to its bare value ${\nu }_l$) in dependence of temperature and wave number, approaching the critical point from above. Finite-size effects, leading to a weaker-than-expected divergence of ${\nu }_{lR}$, are noticeable at low $k$ (see text). The lines are drawn as a guide to the eye.

Figure 6

(a) Critical fluctuation contribution to the shear viscosity in dependence of the reduced temperature. The data (filled symbols) is obtained from simulations using the Green-Kubo relation (70) at $k=0$. The solid curve represents the theoretical prediction of Eq. (68). (b) Wave-number dependence of the critical shear viscosity (at $\theta =0$) obtained from Eq. (70), plotted in double-logarithmic representation. For comparison, the same data is plotted in the inset in log-linear scale. (c) Finite-size behavior of the shear stress correlation function (normalized by the system volume $V=L^2$) and the critical fluctuation contribution to the shear viscosity, ${\nu }_{s,\text{crit}}$ (inset). The dashed curve is $\propto L^{-0.15}$. Time is expressed in units of the characteristic order-parameter relaxation time $1/{\Gamma }_{\mathrm{op}}$. The theoretically predicted power-law divergence of the shear viscosity is not recovered by the present simulations. See text for further discussion.

Figure 7

Schematic diagrams of the additional contributions to the order-parameter self-energy [cf. Eq. (47)] at two-loop order. Unlabeled dots represent the possible couplings $r$, $\kappa$, ${\nu }_l$, ${\nu }_t$ associated with the three-point vertices. Open circles and arrows (indicating correlation or response functions) are understood to be present on some of the internal lines.