Second sound near lambda transition in presence of quantum vortices

Abstract

In this paper, temperature waves (also known as second sound) are considered, with their respective coupling with waves in the order parameter describing the transition from normal phase to superfluid phase, and with waves in the vortex length density. We analyze the coupling between these three kinds of waves and explore its relevance in situations not far from the lambda transition. In particular, the expressions for the second sound speed and second sound attenuation are explicitly obtained within some approximations, showing the influence of the order parameter and the vortex length density, which is decisive close to the transition.

Appendix

Mathematical derivation of field Eq. (2) We outline in this “Appendix” the mathematical derivation of system (2). We start considering the following set of balance equations for the order density $F=\rho f$, the energy density $E=\rho \epsilon$, the heat flux density $\mathbf{q}$ and the vortex line density L:

$$\begin{aligned} \left\{ \begin{array}{l} {\partial _t F}({\mathbf {x}},t)+ \nabla \cdot {\mathbf {J}}^{F}({\mathbf {x}},t)=\sigma ^{F}({\mathbf {x}},t), \\ \partial _t E({\mathbf {x}},t)+ \nabla \cdot {\mathbf {q}}({\mathbf {x}},t) =0, \\ {\partial _t{\mathbf {q}}}({\mathbf {x}},t)+\nabla \cdot {\mathbf {J}}^{{\mathbf {q}}}({\mathbf {x}},t)= \mathbf {\sigma ^{q}}({\mathbf {x}},t), \\ {\partial _t L}({\mathbf {x}},t) + \nabla \cdot \mathbf{J}^L({\mathbf {x}},t) =\sigma ^L({\mathbf {x}},t), \end{array}\right. \end{aligned}$$

(A.1)

where ${\mathbf {J}}^{F}$, $\mathbf {J^{q}}$ and $\mathbf{J}^L$ are the fluxes of the order parameter density, of the heat flux density, and of vortex lines density respectively, while $\sigma ^{F}$, $\mathbf {\sigma ^{q}}$ and $\sigma ^L$ are source terms.

Restrictions on the fluxes are obtained by using the second law of thermodynamics, that states that there exists a convex function $S = S(F,E,L,q^2)$, the entropy density, and a vector function ${\mathbf {J}}^{S}= \phi (F,E,L,q^2) \mathbf{q}$, the entropy flux density, such that, for every thermodynamic process:

$$\begin{aligned} \partial _t S+ \nabla \cdot {\mathbf {J}}^{S}\ge 0. \end{aligned}$$

(A.2)

We will choose for the fluxes the following constitutive equations, linear in the heat flux $\mathbf q$, local in space-time and compatible with the material objectivity principle:

$$\begin{aligned}&{\mathbf {J}}^{F} =\gamma _1(F,E,L) {\mathbf {q}}, \end{aligned}$$

(A.3)

$$\begin{aligned}&\mathbf{J}^L = \nu _1(F,E,L) \mathbf{q}, \end{aligned}$$

(A.4)

$$\begin{aligned}&\mathbf{J ^q} = \beta _1(F,E,L) \mathbf{U}. \end{aligned}$$

(A.5)

We will exploit the consequences of second law using the Liu method of Lagrange multipliers [10]. We get the following inequality, which must be satisfied for arbitrary values of the field variables:

$$\begin{aligned}&\partial _t S +\nabla \cdot {\mathbf {J}}^S -\varLambda ^F \left[ \partial _t F+ \nabla \cdot {\mathbf {J}}^{F} -\sigma ^{F} \right] -\varLambda ^E \left[ \partial _t E +\nabla \cdot \mathbf{q}\right] \nonumber \\&\quad -\, \varvec{\varLambda }^\mathbf{q} \cdot \left[ {\partial _t\mathbf{q}} + \nabla \cdot \mathbf{J}^\mathbf{q} -\sigma ^\mathbf{q} \right] - \varLambda ^L \left[ {\partial _t L} + \nabla \cdot \mathbf{J}^L -\sigma ^L \right] \ge 0, \end{aligned}$$

(A.6)

where $\varLambda ^{F}$, $\varLambda ^{E}$, $\varLambda ^{L}$ and ${\varvec{\varLambda }}^{\mathbf {q}}$ are the so-called Lagrange multipliers.

Imposing the coefficients of time derivatives of the fundamental fields to be zero, we obtain:

$$\begin{aligned} dS=\varLambda ^F dF + \varLambda ^E d E+\varLambda ^LdL + {\varvec{\varLambda }}^{\mathbf {q}}\cdot d{\mathbf {q}}. \end{aligned}$$

(A.7)

We introduce now the non-equilibrium temperature $\theta$ of liquid helium (in the normal and in the superfluid phase), that we assume, by analogy with classical thermodynamics, given by [2]:

$$\begin{aligned} \frac{1}{\theta }=\left[ {\frac{\partial S}{\partial E}}\right] _{F,L, {\mathbf {q}}}. \end{aligned}$$

(A.8)

In this paper, we choose the temperature $\theta$ (that is the parameter that controls the superfluid transition) as independent variable, instead of the energy density E. Therefore, we introduce the free energy density $G=E-\theta S$, that is a function of fields F, $\theta$, L and $q^2$. From (A.7) we get:

$$\begin{aligned} dG= -S d\theta -\theta \varLambda ^F dF -\theta \varLambda ^Ld L -\theta {\varvec{\varLambda }}^{\mathbf {q}} \cdot d{\mathbf {q}}. \end{aligned}$$

(A.9)

Following the Ginzburg–Landau (GL) general theory of second-order phase transitions, we will suppose that the free-energy density G can be expanded in powers of $F^2$ in the following way:

$$\begin{aligned} G = G_{1} +G^{F}=G_{1}+\frac{1}{2}\tilde{a} F^2+\frac{1}{4}\tilde{b} F^4, \end{aligned}$$

(A.10)

where $G_{1}$ is the free-energy density of He I, and (see [1, 19]):

$$\begin{aligned} \tilde{a}= {{\mathcal {A}}}(L,q^{2}) (\theta - \theta _c) \quad {\mathrm{and}}\quad {\tilde{b}}= \frac{{\mathcal {A}}\theta _{c}}{\rho ^{2}}, \end{aligned}$$

(A.11)

where $\theta _{c}=\theta _{c}(p,q^2)$ is the transition temperature.

To write the evolution equation for F, following Landau, we assume that the stable steady solution of it minimizes the total free energy with respect to F. Therefore, we assume that the production term in the equation of the order parameter is proportional to ${\delta G}/{\delta F}$. We get:

$$\begin{aligned} \sigma ^{F}=-K_f\frac{\delta G}{\delta F}, \end{aligned}$$

(A.12)

where $K_f$ is a positive coefficient. Then we obtain the following evolution equation for the phase field F:

$$\begin{aligned} {\partial _t F}+\nabla \cdot \left( \gamma _1\mathbf{q} \right) =-K_f{\mathcal {A}}\theta _c \left[ \frac{\theta }{\theta _c}-1+\frac{F^2}{\rho ^2}\right] F. \end{aligned}$$

(A.13)

Note that the dimension of $K_f{{{\mathcal {A}}}} \theta _c$ is the inverse of a time. So, we can introduce a relaxation time $\tau _f$ for the order parameter f, as:

$$\begin{aligned} \frac{1}{\tau _f(L,q^2)} =K_f{{{\mathcal {A}}}} \theta _c. \end{aligned}$$

(A.14)

We obtain the following GL-type equation for F:

$$\begin{aligned} {\partial _t F}+\nabla \cdot \left( \gamma _1\mathbf{q} \right) =-\frac{1}{\tau _f(L,q^2)} \left[ \frac{\theta }{\theta _c}-1+\frac{F^2}{\rho ^2}\right] F. \end{aligned}$$

(A.15)

Finally, from (5) it results:

$$\begin{aligned}&\varLambda ^{F} = -\frac{1}{\theta }\left[ \frac{\partial G}{\partial F }\right] _{\theta ,L, {\mathbf {q}}}= - {\mathcal {A}} F \left[ 1-\left( 1-\frac{F^2}{\rho ^2} \right) \frac{\theta _{c}}{\theta } \right] , \end{aligned}$$

(A.16)

$$\begin{aligned}&\varLambda ^{L } = -\frac{1}{\theta }\left[ \frac{\partial G}{\partial L}\right] _{\theta ,F, {\mathbf {q}}}= - \frac{\partial {\mathcal {A}}}{\partial L} \frac{F^2}{2} \left[ 1-\left( 1-\frac{F^2}{2\rho ^2} \right) \frac{\theta _{c}}{\theta } \right] , \end{aligned}$$

(A.17)

$$\begin{aligned}&{\varvec{\varLambda }^\mathbf{q}} = -\frac{1}{\theta }\left[ \frac{\partial G}{\partial {\mathbf {q}}}\right] _{\theta ,F, L}= - {F^2} \left[ \frac{\partial {\mathcal {A}}}{\partial q^2} + \frac{1}{\theta }\left( 1-\frac{F^2}{2\rho ^2} \right) \frac{\partial ({\mathcal {A}}\theta _{c})}{\partial q^2} \right] {\mathbf {q}}. \end{aligned}$$

(A.18)

We determine now the consequences for the fluxes given by the second of thermodynamics. Putting ${\varvec{\varLambda }^\mathbf{q}}= \lambda {\mathbf {q}}$ and imposing the coefficients of space derivatives to be zero in Liu inequality (A.6), we obtain:

$$\begin{aligned} \nabla \cdot {\mathbf {J}}^S -\varLambda ^F \nabla \cdot {\mathbf {J}}^{F} -\varLambda ^E \nabla \cdot \mathbf{q} - \lambda \mathbf{q} \cdot \nabla \cdot \mathbf{J}^\mathbf{q} - \varLambda ^L \nabla \cdot \mathbf{J}^L = 0. \end{aligned}$$

(A.19)

From which we get:

$$\begin{aligned}&\phi - \varLambda ^E -\varLambda ^F \gamma _1 -\varLambda ^L \nu _1 = 0, \end{aligned}$$

(A.20)

$$\begin{aligned}&d\phi = \lambda d\beta _1 +\varLambda ^F d \gamma _1 +\varLambda ^L d \nu _1. \end{aligned}$$

(A.21)

By using these equations, one obtains the expression for $d \beta _1$:

$$\begin{aligned} \lambda d \beta _1 = d ( \theta ^{-1} ) + \gamma _1 d \varLambda ^F + \nu _1 d\varLambda ^L. \end{aligned}$$

(A.22)

With the following positions:

$$\begin{aligned}&{\overline{\zeta }} = \frac{\partial \beta _1 }{\partial \theta }=-\frac{1}{\lambda \theta ^{2}}+\frac{1}{ \lambda } \left[ \gamma _1\frac{\partial \varLambda ^{F}}{\partial \theta } + \nu _1\frac{\partial \varLambda ^{L}}{\partial \theta } \right] , \end{aligned}$$

(A.23)

$$\begin{aligned}&{\overline{\varphi }}_1 = \frac{\partial \beta _1}{ \partial L} =\frac{1}{ \lambda } \left[ \gamma _1\frac{\partial \varLambda ^{F}}{\partial L} + \nu _1\frac{\partial \varLambda ^{L}}{\partial L} \right] , \end{aligned}$$

(A.24)

$$\begin{aligned} {\overline{\varphi }}_2= & {} \frac{\partial \beta _1 }{\partial F }=\frac{1}{\lambda }\left[ \gamma _1\frac{\partial \varLambda ^{F}}{\partial F} + \nu _1\frac{\partial \varLambda ^{L}}{\partial F} \right] , \end{aligned}$$

(A.25)

Equation (A.22) is written:

$$\begin{aligned} d\beta _1 ={\overline{\zeta }} d\theta + {\overline{\varphi }}_1 dL+{\overline{\varphi }}_2 dF. \end{aligned}$$

(A.26)

The heat flux equation For the production terms $\mathbf {\sigma ^{q}}$ in the equation for the heat flux, we will choose the expression:

$$\begin{aligned} \sigma ^\mathbf {{q}}=-\frac{1}{\tau _{q}(F,L)}{\mathbf {q}}, \end{aligned}$$

(A.27)

where $\tau _{q}(F,L)$ can be interpreted as relaxation time. A possible expression for $\tau _q$ could be [1]:

$$\begin{aligned} \tau _{q}(F,L)= \tau _0 \, \frac{\frac{F^2}{\rho ^2}}{1-\frac{F^2}{\rho ^2}(1-\tau _0 K L)}= \tau _0 \, \frac{{F^2}}{\rho ^2- {F^2} (1-\tau _0 K L)}, \end{aligned}$$

(A.28)

where $\tau _{0}$ is a constant having the dimension of time. In this way and using (A.26), the evolution Eq. (A.1c) for the heat flux becomes

$$\begin{aligned} \frac{F^2 }{\rho ^{2}} \left\{ \partial _t {\mathbf {q}}+ \left[ {\overline{\zeta }}\nabla \theta + {\overline{\varphi }}_1 \nabla L+ {\overline{\varphi }}_2\nabla F \right] \right\} = - \left( \frac{\rho ^2-F^2}{\tau _0 \rho ^2} +\frac{F^2}{\rho ^2} K L \right) {\mathbf {q}}. \end{aligned}$$

(A.29)

The vortex line density evolution equation A simple expression for the source term $\sigma ^L$ in line density evolution equation, is:

$$\begin{aligned} {\sigma ^{L}}= \alpha _1 L^{{3}/{2}} |\mathbf{q}| -\frac{\rho ^2}{F^2} \beta \kappa L^2, \end{aligned}$$

(A.30)

in this way the evolution equation for the vortex line density (A.1d) can be written:

$$\begin{aligned} \frac{F^2}{\rho ^2}\left[ \partial _t L + \nabla \cdot \left( \gamma \mathbf{q} \right) \right] = \frac{F^2}{\rho ^2} \alpha _1 L^{{3}/{2}} |\mathbf{q}| - \beta \kappa L^2. \end{aligned}$$

(A.31)

When $F=0$, i.e. in the normal phase, it results $L=0$, while, when $F=\rho$, i.e. sufficiently below the superfluid transition, we obtain a generalization of the well known Vinen equation to inhomogeneous situations [22].

Field equations Substituting the expressions of the production terms $\sigma ^F$, $\sigma ^L$ and $\sigma ^{{\mathbf {q}}}$ obtained in (A.12), (A.27) and (A.30) in Eq. (A.1), the following system of field equations is written:

$$\begin{aligned} \left\{ \begin{array}{l} {\partial _t F}+\nabla \cdot \left( \gamma _1\mathbf{q} \right) =\sigma ^F= -\frac{1}{\tau _f(L,q^2)} \left[ \frac{\theta }{\theta _c}-1+\frac{F^2}{\rho ^2}\right] F, \\ \partial _t {\mathbf {q}}+ \left[ {\overline{\zeta }}\nabla \theta + {\overline{\varphi }}_1 \nabla L+ {\overline{\varphi }}_2\nabla F \right] =- \left( \frac{\rho ^2-F^2}{\tau _0 F^2} +K L \right) {\mathbf {q}},\\ \partial _t L +\nabla \cdot \left[ \nu _1\mathbf{q} \right] =\sigma ^L= \alpha _1 L^{{3}/{2}} |\mathbf{q}| - \frac{\rho ^2 }{F^{2}} \beta \kappa L^2, \\ \rho c \partial _t \theta +\nabla \cdot {\mathbf {q}} = - \rho \frac{\partial \epsilon }{\partial F} \partial _t F - \rho \frac{\partial \epsilon }{\partial L} \partial _t L - 2\rho \frac{\partial \epsilon }{\partial q^2}{\mathbf {q}} \cdot \partial _t {\mathbf {q}}, \end{array}\right. \end{aligned}$$

(A.32)

where $\epsilon$ is the specific internal energy of helium and c the specific heat.

This system can be easily identified with system (2), if we neglect the dependence of the specific energy by the heat flux (that would lead to non-local and non-linear terms), we recall that we have assumed constant the mass density, and we will make the positions:

$$\begin{aligned}&\zeta ^*= f^2 {\overline{\zeta }}, \varphi _1={\overline{\varphi }}_1, \varphi _2=\rho _0 {\overline{\varphi }}_2, f^2 \gamma = \frac{{\gamma }_1}{\rho _0}, \\&\nu =\nu _1, \epsilon _1=\rho \frac{\partial \epsilon }{\partial L}, \epsilon _2=\rho \frac{\partial \epsilon }{\partial f}, \end{aligned}$$

whose physical meaning is explained in Sect. 2.