Entropy current for non-relativistic fluid

We study transport properties of a parity-odd, non-relativistic charged fluid in presence of background electric and magnetic fields. To obtain stress tensor and charged current for the non-relativistic system we start with the most generic relativistic fluid, living in one higher dimension and reduce the constituent equations along the light-cone direction. We also reduce the equation satisfied by the entropy current of the relativistic theory and obtain a consistent entropy current for the non-relativistic system (we call it"canonical form"of the entropy current). Demanding that the non-relativistic fluid satisfies the second law of thermodynamics we impose constraints on various first order transport coefficients. For parity even fluid, this is straight forward; it tells us positive definiteness of different transport coefficients like viscosity, thermal conductivity, electric conductivity etc. However for parity-odd fluid, canonical form of the entropy current fails to confirm the second law of thermodynamics. Therefore, we need to add two parity-odd vectors to the entropy current with arbitrary coefficients. Upon demanding the validity of second law, we see that one can fix these two coefficients exactly.


I. INTRODUCTION AND SUMMARY
Hydrodynamics is an effective description of nearly equilibrium interacting many body system. A fluid system is considered to be continuous, i.e. when we talk about a small volume element (or 'fluid particle') of the fluid, it still contains a large number of constituent particles (atoms/molecules). More specifically, the size of the fluid particle is much much greater than the mean free path of the system. The equations of hydrodynamics assume that the fluid is in local thermodynamic equilibrium at each point in space and time, even though different thermodynamic quantities like fluid velocity v( x, t), energy density ǫ( x, t), pressure p( x, t), fluid density ρ( x, t), temperature τ ( x, t) etc. may vary. Fluid mechanics applies only when the length scales of variation of thermodynamic variables are large compared to equilibration length scale of the fluid, namely the mean free path [1].
Although hydrodynamics is an old and well-studied subject in physics, recently there has been much interest and progress in relativistic, charged, viscous fluid in presence of some global anomaly. The first evidence of relativistic anomalous fluids was observed in [2,3], in the context of the AdS/CF T correspondence. In these papers the authors found a new term (and hence a new transport coefficient) in the charge current in presence of a Chern-Simons term in the bulk Lagrangian. Later, in [4] it has been shown that this kind of term in charge current is not only allowed by the symmetry but is required by triangle anomalies and the second law of thermodynamics. Demanding the positivity of local entropy current they showed that the coefficient of this new term is also fixed in terms of anomaly coefficient of the theory. In general, the second law of thermodynamics (or equivalently, positivity of local entropy current) imposes restrictions on different transport coefficients. Interested readers can look at [5][6][7][8][9][10][11][12].
This was about relativistic fluids. Much attention has not been paid in parity-odd charged non-relativistic fluid in presence of background electromagnetic fields. In [13] an attempt has been made to study non-relativistic neutral fluids as a consistent light-cone reduction of relativistic fluid systems. In [14], the idea has been extended to charged fluid without any background field. However, main focus of these two papers was to construct the holographic duals of long wave length fluctuations (hydrodynamic limit) of conformal non-relativistic field theories. They also computed few transport coefficients (for example thermal conductivity) holographically.
Recently in [15], non-relativistic, parity-odd, first order charged fluid in 2 dimensions has been discussed. The authors started with a 2 + 1 dimensional relativistic, parity-odd charged fluid in presence of background gauge fields and took suitable non-relativistic limit of the system. They identified the parity-violating contributions to the non-relativistic constitutive relations, which include the Hall current flowing perpendicular to the temperature gradient, the Hall viscosity and the Leduc-Righi energy current 1 . However, in these papers, much attention has not been paid to second law of thermodynamics. It is well known that the local second law of thermodynamics imposes several constraints on the flow and the transport coefficients. For example, the local second law demands that shear, bulk viscosity coefficients and thermal conductivity have to be positive definite for a non-relativistic fluid [1]. This particular issue, for relativistic fluid, has been discussed vividly in various papers mentioned above. However, the constraints from second law of thermodynamics are not yet well understood for a general class of non-relativistic fluids (charged fluid in presence of background fields). Goal of this paper is to write down a local entropy current, which satisfies second law of thermodynamics, for a non-relativistic charged fluid in presence of arbitrary background electromagnetic field and to study the constraints on the transport coefficients.
One way to obtain the energy momentum tensor, charge current and entropy current for a non-relativistic theory is to reduce the constitutive equations and entropy current equation of a relativistic fluid over a light-cone direction. We start with a (3 + 1)-dimensional 1 See also [16]. relativistic charged fluid with global U (1) 3 anomaly and write down their non-relativistic counterparts. We follow the light-cone reduction technique, which was introduced in [13], and shall as well review it in our paper. Let us conclude this section with an outline and summary of our main results.
Outline and Summary : • In Section (II) we consider first order, relativistic, charged anomalous fluid in (3 + 1) dimensions in presence of background electromagnetic fields. We briefly review the work of [4] and show how parityodd transport coefficients are related to the coefficient of anomaly from the positivity of divergence of local entropy current.
• In Section (III) we highlight the important equations of non-relativistic hydrodynamic system.
• In Section (IV) we consistently obtain the energy current density (j i ), stress tensor (t ij ) and charge current density (j i I ) of non-relativistic fluid by Light Cone Reduction (LCR). We also find different non-relativistic fluid variables (thermodynamic quantities and transport coefficients) in terms of the relativistic fluid variables. Here we mention the final expression for stress tensor, energy current and charge current: All the non-relativistic transport coefficients are fixed in terms of relativistic transport coefficients and other thermodynamic variables. Explicit relations have been provided in the main text.
• In charge current we identify the thermal Hall contributionκ I ǫ ij ∇ j τ (where τ is the temperature of non-relativistic fluid). The Hall energy flow sourced by the temperature gradient is known as the Leduc-Righi effect. It has been predicted in condensed matter physics, that this kind of effect can be observed in various topological insulators.
• We also obtain the electromagnetic Hall energy current (parity-odd)σ IJ ǫ ij ǫ Jj − v k β Jkj , where ǫ Jj is the applied electric field and β Iij is proportional to the applied magnetic field (check Eqn. (III.49) for exact definitions).
• We find that the non-relativistic fluid system satisfies the famous 'Wiedemann-Franz Law'.
• Finally, we compute the entropy current j i S for the non-relativistic fluid in Section (V), and demand that the rate of entropy production of the system is always positive definite. For parity-even fluid it turns out to be quite straight forward. The canonical form of the entropy current confirms that the total entropy always increases due to dissipation, thermal and charge conduction. However for parity-odd fluid, canonical form of the entropy current fails to confirm the second law of thermodynamics. Therefore, we need to add two parity-odd vectors to the entropy current with arbitrary coefficients. Upon demanding the validity of second law, we see that one can fix these two coefficients exactly.
• Finally we observe that, unless the fluid is incompressible, all the parity-odd transport coefficients must vanish to satisfy the second law of thermodynamics.

II. RELATIVISTIC ANOMALOUS CHARGED FLUID
Hydrodynamic description does not follow from the action principle, rather it is generally formulated in the language of equations of motion. The reason for this is the presence of dissipation in thermal media. Due to internal friction called viscosity, a dissipative fluid loses its energy over time as it propagates. The fluid without any viscous drag is called an ideal fluid. In the simplest case, the hydrodynamic equations (for a uncharged fluid) are the laws of conservation of energy and momentum, A relativistic fluid in (d + 1, 1) dimensions (d + 1 spatial and 1 time dimensions) has total d + 3 independent intensive fluid variables: temperature (T ), pressure (P ) and velocities u µ (only d + 1 components of the velocity are independent due to normalization u µ u µ = −1). Extensive variables energy density (E) and entropy density (S) are not considered to be independent as they can be determined in terms of the intensive variables, by the first law of thermodynamics and the equation of state. Additionally we have an Euler's relation which follows from the extensivity of the internal energy: which in conjunction with the first law relates P and T as: Therefore, among these thermodynamic variables, we can consider only one to be independent, which we choose to be T . Thus, fluid is determined in terms of d + 2 variables: u µ , T . On the other hand we have d + 2 constitutive equations (II.2), and hence fluid system is completely determined. We can express the energy momentum tensor T µν as a function of temperature, velocity and their derivatives.
In case of fluid with multiple conserved U (1) currents, we have more fluid variables: charge densities (Q I ).
The corresponding conjugate variables are called chemical potentials denoted by (M I ). In presence of conserved charges the first law is modified to From this equation one can compute the chemical potentials of the fluid in terms of charges and other variables. However, in our computation we consider chemical potentials M I to be our independent variables, not the charges. As a result, number of fluid variables have been increased to d + 3. However, in this case we also have another constitutive equation (which follows from the global U (1) symmetries), Euler's relations are also modified to: and from the first law one can write, Therefore, for charged fluid we consider our basic thermodynamic variables to be T and M I /T , rest are fixed in terms of these two variables. We can therefore write the last equation in the following form: Thus, for a charged fluid, we express energy-momentum tensor and charge current as a function of fluid velocities, temperature, chemical potential and their derivatives.
One striking feature of a relativistic quantum field theory is triangle anomalies. The effect of this anomaly is reflected in three point correlation function of charge currents. However, the anomaly does not affect the conservation of the current associated with a global symmetry. When we put the theory in external background gauge fields coupled to the currents, some of the currents will no longer be conserved.
In the next subsection we discuss about the form of the energy momentum tensor and global U (1) current of a relativistic anomalous fluid in presence of background gauge fields.
A. Anomalous charged fluid: energy-momentum tensor and current We review a generic charged relativistic fluid in (d+2)dim with anomalies. The energy-momentum tensor is given by: where, and fluid velocities are normalized Here, η and ζ are relativistic shear and bulk viscosity coefficients respectively. Up to first order in derivatives, the respective terms are given by: where we use the projection operator: Similarly charge current of a relativistic fluid in presence of multiple charges Q I (I = 1, 2, 3) is given by 2 where, Here ̺ IJ , λ IJ and γ I are charge, electric and thermal 3 conductivities respectively. However, demanding the positivity of local entropy current one can show show that, γ I vanishes and ̺ IJ is related to λ IJ . We have kept the fluid in some background electromagnetic gauge fields given by A µ I : for a beautiful review. 3 Thermal conductivity of relativistic theory is fundamentally different from the thermal conductivity of non-relativistic theory. Entropy positivity demands relativistic thermal conductivity to vanish, however in non-relativistic limit it origins from elsewhere.
The last two terms of Eqn. (II.17) are the most generic single-derivative parity-odd modifications to the charge current allowed in Landau frame. These terms are specific to (3 + 1)-dimensions. In higher dimensions, the parity odd terms appear only at higher derivative orders. However, we would like to keep our calculations generic to (d + 2)-dim, therefore in all expressions, we represent parity-odd terms in curly braces {. . .}, with an understanding that these terms contribute only for d = 2.
We are working in Landau frame, which demands and hence: The constitutive equations for this fluid are given by: C IJK is called anomaly coefficient, which is completely symmetric in all indices.
Leading order T µν and J µ must obey the first order conservation equations, which can be found trivially using Eqn. (II.23), (II.24): We have projected the equations along and perpendicular to the direction of velocity for later convenience. We have also used E µ I u µ = 0, which can be seen trivially. Using these, τ µν from Eqn. (II.13) can be written as: where, The trace of the energy-momentum tensor is given by: For conformal fluids T µ µ = 0, which can be reached by setting T = ζ = 0.
However the last two terms of Eqn. (II.37) cannot be made positive definite. One would then expect that coefficients of those terms must vanish! But in [4], the authors identified that Eqn (II.36) is not the most generic expression for the entropy current. They modified the entropy current with the most generic parity odd vectors allowed by the symmetry In presence of these two terms the parity-odd part of the last equation becomes, Demanding the positivity of divergence of local entropy current we get, One can in principle solve the above set of equation to get relations between D,D I , ℧ I ,℧ IJ and C IJK .
Finally we have the desired form of entropy current which is entirely positive definite.

III. NON-RELATIVISTIC CHARGED FLUID
In this section, we review the properties of charged non-relativistic dissipative fluid living in (d+1)dimensions, in presence of some background gauge fields.
A non-relativistic fluid has following constitutive equations: Continuity equation: where ρ is density of fluid particle.
Momentum conservation equation: where t ij is the energy-momentum tensor.
Energy conservation equation: where j i is the energy current.
Conservation of charge current: where, q I 's are the U (1) charge densities and j I 's are the corresponding currents. Note that energy current is denoted by j where as charge current is denoted by j I .
In d spatial dimensions, charged fluid has total d + 3 independent variables (pressure p, temperature τ , chemical potentials µ I and velocity v), while others (density ρ, energy density ǫ, charge density q I and entropy density s) are fixed in terms of these using thermodynamic relations. Unlike the relativistic case we do not consider the Euler's relation here. Later we shall see that the lightcone reduction does not preserve extensivity of the fluid system. Hence, the fluid is determined by d + 3 parameters chosen to be: τ , µ I /τ , p and v (same as (d+1,1)-dim relativistic fluid). On the other hand we have d + 3 constitutive equations.
Therefore, we can express energy-momentum tensor and charge current in terms of τ , µ I /τ , p, v and their derivatives. For ideal fluid they are given by, A. Non-relativistic dissipative fluid in background electromagnetic field When we put a dissipative fluid in background electromagnetic fields, the above equations are modified. Continuity equation and charge conservation equation remain the same but other two equations are modified to where, are electric and magnetic fields respectively. φ I and a i I are scalar and vector potentials respectively. For dissipative fluid the stress-energy tensor is given by, where π ij is the correction to ideal fluid stress tensor due to dissipation n and z are the non-relativistic shear and bulk viscosity coefficients respectively. The energy current for dissipative fluid is given by, where, (III.54) Here κ, ξ I , σ I and α I are thermal, charge, electric and magnetic conductivities respectively. Later we shall see that, for non-relativistic fluid, obtained by light cone reduction of a relativistic fluid, ξ I and α I are related to σ I .
Similarly charge current for a dissipative fluid is given by, where, κ I ,σ IJ ,ξ IJ ,m I ,α IJ andκ I ,σ IJ ,ξ IJ ,m I ,ᾱ IJ are some arbitrary parity-even and parity-odd transport coefficients 4 . Again, for a non-relativistic fluid, obtained by light-cone reduction these transport coefficients are fixed in terms of relativistic transport coefficients. Also, they are constrained when we demand positivity of local entropy current.

IV. LIGHT-CONE REDUCTION
Discrete light cone quantization of a (d+1, 1)-dim relativistic quantum field theory (QFT) boils down to a nonrelativistic quantum field theory in one lower dimension. Let us consider a QFT in flat spacetime with metric, (IV.57) We introduce the light-cone coordinates, In the light-cone frame the metric can be written as, Suppose we view the QFT in this light-cone frame and evolve it in light-cone time x + . Then, for fixed lightcone momentum P − , we obtain a system in d + 1 dimensions with non-relativistic invariance. This is because the symmetry algebra of the relativistic theory reduces to corresponding non-relativistic symmetry algebra upon light-cone reduction. For example, if the relativistic theory is invariant under Poincaré transformation, then the corresponding algebra (SO(d + 1, 1)) reduces to d dimensional Galilean algebra. On the other hand if we consider a QFT with conformal invariance in (d + 1) + 1 dimensions, then upon light cone reduction the corresponding algebra (SO(d + 2, 2)) reduces to Schrödinger algebra in d spatial dimensions. This is known as discrete light-cone quantization of quantum field theories. Since hydrodynamics is low energy fluctuation of equilibrium quantum field theory, light-cone reduction of relativistic constitutive equations boil down to the nonrelativistic constitutive equations for a fluid in one lower dimension. Relativistic charged fluid in (d + 1, 1)-dim, as we have already discussed, has d + 3 degrees of freedom: temperature, chemical potential and normalized velocities. On the other had, a non-relativistic fluid in d spatial dimensions also has total d + 3 degrees of freedom: temperature, pressure, chemical potential and velocities. Our goal is to consider the most generic relativistic anomalous fluid system in presence of background electromagnetic fields in (3 + 1) dimensions and reduce the constitutive equation (also equation for entropy current) over light-cone coordinates and obtain the corresponding non-relativistic equations (most generic) for a fluid in one lower dimensions. We also find a mapping between the degrees of freedom of the (d+2)-dimensional fluid to the degrees of freedom of the (d + 1)-dimensional fluid.
We denote the d spatial coordinates with x i . Metric components in light-cone coordinates are given by g ij = δ ij and g +− = −1, (IV.60) rest are zero. Gradient operator is given by, (IV.61) As we have seen that not all components of u µ , J µ I and T µν are independent, from Eq. (II.22) we get, We shall reduce the theory along the x − direction, and consider x + to be the non-relativistic time. We will consider only solutions to the relativistic equations that do not depend on x − ; that is, all derivatives ∂ − vanish.

A. Reduction of background fields
We reduce the Maxwell's equations for the background fields of relativistic fluid to get consistent background for non-relativistic theory. Maxwell's equations for relativistic system are given by (we shall assume that the sources for these background fields are far away from the fluid), Under light-cone reduction the above equations take the following form: These equations can be identified with source free static Maxwell's equations of a non-relativistic system if we map 5 : and Which would inturn tell us We would however like A + I to have some finite value at infinity, which will enforce: With this identification, from Eq. (II.20) one can show that 71) and In the light-cone coordinates it becomes Thus we see that the gauge freedom of relativistic theory reduces to the gauge freedom of the non-relativistic theory. Additionally A + , which we have fixed to be a constant, under a gauge choice does not change. Otherwise, one could perform a local gauge transformation to modify A + to a non constant value, which would then break our identification.

B. Reduction of energy-momentum tensor and charge current
The reduction of relativistic equations of energymomentum and charge conservation after using Eqn. (IV.70) are given by: These equations reduce to non-relativistic equations under following identifications The identification also tells that the non-relativistic charge current is conserved, while the relativistic charge current was not. We shall now attempt to use this mapping to establish relations between relativistic and nonrelativistic parameters.

T i+ calculation
Using Eqn. (II.28) and (II.14) we can show: where, (IV.84) Using Eqn. (IV.82), we will find, Using Eqn. (II.28) we will find: It's easy to check that: and thus introducing Z from Eqn. (II.29), Using the above relations and the identity: we find that: (IV.91) where σ ij is given by Eqn. (III.52). Similarly using Eqn. (II.14) we can reach to: and thus: (IV.99) The last term can be written as: We identify (leading order): With this identification we get, Comparing it with (III.54), we find that we have the expected form. Also we have estabilished relations between charge, electric and magnetic conductivities: Using Eqn. (II.17) we can find Using Eqn. (IV.72) we can write Identifying J + I = q I , where, This tells us the sub-leading correction to non-relativistic charges.

J i calculation
Using Eqn. (II.17) we can find One can find using Eqn. (IV.100) that: Certain velocity dependent terms appear in the above expression, which can be shown to sum to zero in 2 spatial dimensions.
Using thermodynamic relations Eqn. (II.7), (II.8), (IV.100), along with all light cone identifications, we will find (IV.114) We identifỹ where, Finally we have the desired form of charge current Note that all transport coefficients are not independent. The only independent coefficients are:m I ,ξ IJ ,σ IJ , ω I andω IJ .
Wiedemann-Franz Law: This famous law predicts that the ratio of charge conductivity (which appears in charge current) 6 to thermal conductivity (which appears in energy current) in metals as:σ/κ = 1/Lτ , where L is the Lorenz number predicted to be ∼ 2.45 × 10 −8 W ΩK −2 . The law is found to be in good agreement with experiments. We attempt to check the same in our setup 7 :σ We model the electrons in metals as free classical gas with no external pressure: fluid with homogeneous particles each of charge e (electronic charge), mass m e and average energy 3/2k B τ . One can check that under mentioned assumptions, our system follows Wiedemann-Franz Law with Lorenz number given as: Assuming ̺ and n of nearly same order, first term turns out to be about 15 orders of magnitude smaller than the second term and can be safely neglected. Hence Lorenz number is our case is given approximately by: 6.68 × 10 −8 W ΩK −2 , given our assumptions, which is in fair agreement with the experimental value.
C. Thermodynamics of the reduced system We start with a relativistic system with variables satisfying the Eq. (II.5) and (II.7). The first equation is the first law of thermodynamics and the second equation is called the Euler's equation. The Euler equation follows from the additive property of internal energy which is a homogeneous function of degree one. After light-cone reduction when we map non-relativistic variables in terms of relativistic ones, we see that at the leading order (ideal fluid), non-relativistic variables satisfy the following two equations, Here we have identified the non-relativistic entropy density s = Su + . The first equation, which follows from Eq. (II.5), is the first law of thermodynamics satisfied by the non-relativistic variables. The second, coming from Eq. (II.7), is not exactly the Euler's equation for nonrelativistic variables. We comment on this equation at the end of this sub-section.
As we consider the first derivative corrections to nonrelativistic variables (evaluated in the previous subsection), the first law of thermodynamics changes. However, that can not be the case. If we demand that our non-relativistic system is a physical one, then the first law should be satisfied by the non-relativistic variables at every order in derivative expansion. This demand forces us to add some higher derivative corrections to entropy density which we could not determine in Section (IV B). Let's assume the following corrected forms: χ is corresponding corrections. Demanding first law of thermodynamics to be satisfied at first order in derivative we get: Note that, thermodynamic relations (II.7) -(II.8), conservation equations (II.25) -(II.27) up to first order and all the identifications between non-relativistic and relativistic variables (obtained by LCR) have been extensively used to carry out the above computation. Now, using (II.25) -(II.27), we can write the above equation in the following form 8 This equation fixes the correction to the entropy density. The last equation in (IV.121), which follows from Euler's equation (for the relativistic variables) does not imply that the non-relativistic energy density ǫ is a homogeneous function of degree one. In fact, this equation gets modified with the corrected values of non-relativistic entropy density 9 . Therefore, we conclude that the nonrelativistic system we obtain after light-cone reduction of a homogeneous relativistic fluid system, is not homogeneous any more (even at the leading order) i.e. the internal energy is not a homogeneous function of degree one.

V. NON-RELATIVISTIC ENTROPY CURRENT
The goal of this section is to write down an entropy current for a non-relativistic fluid system, and find constraints on various transport coefficients from the positivity of local entropy production. Using Eqn. (IV.124) and properties of Y µν , one can reduce the Eqn. (II.37) and after a lengthy algebra, we get the following equation, where j i S is the non-relativistic canonical entropy current defined as (similar to relativistic canonical entropy current given by Eqn. (II.35)), 8 There will be two terms proportional to C IJ K E µ J B Kµ which we have checked will vanish after reduction. 9 Although the form of this equation can be maintained by adding some one derivative correction to temperature or chemical potential.  We can rewrite Eqn. (V.125) in more conventional form as The first three terms on the right hand side survive for uncharged dissipative fluid [1]. Here we compute the other terms which are responsible for local entropy production for parity-odd charged fluid in presence of background fields. Since we demand that the fluid satisfies the second law of thermodynamics, the right hand side of Eqn (V.125) should be positive definite. This implies following constraints on the transport coefficients These constraints are compatible with what we get from relativistic entropy production condition. However the last term in Eqn. (V.125) is a parity odd term and can not be written as a perfect square. Therefore, it seems that the presence of this term breaks the second law of thermodynamics. However, similar to the relativistic fluid dynamics, we can add extra parity-odd terms (allowed by symmetry) to entropy current to make right hand side a perfect square.
Note that the last term in Eq. (V.125) appears only in 2+1 dimensions. Since the first order parity-odd terms in Eqn. (V.125) do not appear in the non-relativistic fluid theories in more than two spatial dimensions, we have a complete and consistent description 10 for these theories. We summarize the results for them in Tables (I), (II).

A. Parity-odd corrections to NR Entropy Current
For fluids in two spatial dimensions however, we must add some parity-odd terms to the entropy current with some arbitrary transport coefficients and determine them from the condition that there is no local entropy loss during the flow of the system. We make the most generic parity-odd modification to the entropy current in two spatial dimensions as follows a, b I , c, d I and f I are arbitrary transport coefficients. After adding these extra terms the right hand side of Eqn. (V.125) can be written as (parity-odd part only) To obey the second law of thermodynamics we claim that the coefficients of all the independent terms must vanish. This is possible except for the last two terms, for which we get the following sets of constraints: additionally, the following matrices must be symmetric: It should be noted that the new coefficients introduced in j i S need not be most generic, and their only significance is to make the parity-odd terms in entropy current vanish. Hence, a minimal choice is as follows: which solves the respective constraints, and leaves us only two independent coefficients in entropy current: b I and d I . Now we turn attention towards the last two terms of Eqn. (V.130). They vanish if we consider the fluid is incompressible and placed in a time independent background. For compressible fluid the second last term is non-zero and can be negative. Hence the second law implies that the f I = d I = 0. As a result all the parity-odd terms vanish for compressible fluid. The resultant fluid is completely parity-even and we summarize the results in tables (I), (II). On the other hand if we consider the background field is time dependent, then the last term can violate the second law. However, in that case, one has to be careful about the change of entropy of the background also. The second law should hold for the fluid and the background system together. The last term produces some vortex kind of motion in the fluid. In that case the velocity of fluid particles around those vortexes becomes very large and our analysis (derivative expansion) may break down.
It should be noted that we demand our non-relativistic system to obey the second law of thermodynamics. We start with a relativistic fluid but never impose any physicality constraint (except the fluid satisfies first law of thermodynamics) on the relativistic side. Relativistic system, in our case, can be considered as a mathematical model (where we allow all possible terms allowed by the symmetry). After reduction we see that at least for the parity-even sector constraint from the second law turns out to be same (compatible) both in relativistic and nonrelativistic case (Eq. Eqn. (II.38) and (V.128)). But for parity-odd sectors they are completely different. For example, C IJK do not even appear in non-relativistic relations Eqn. (V.131) -(V.133), though for relativistic fluid the second law relates ℧ I and℧ IJ to C IJK (Eq. Eqn. (II.41)). Whereas for non-relativistic fluid one can consider ω I andω IJ to be independent and f I and d i are fixed in terms of them. However, one can also start with a relativistic fluid satisfying second law of thermodynamics (as an additional condition). In that case for an incompressible fluid it is possible to express the nonrelativistic transport coefficients (f I , d I , ω I ,ω I ) in terms of anomaly coefficients of the relativistic system. But for compressible fluid it is not possible as these coefficients turn out to be zero which implies that ∇ k v k = 0, or in other words, there is no compression or expansion of the fluid during the flow. Using the non-relativistic constitutive equations one can easily show that: which means that for incompressible fluids, dǫ/dt is atmost two derivative in order, and is zero for ideal fluids.
Considering this result at first derivative order we can infer that: in first derivative order. Same can be inferred directly from Eqn. (IV.88).
Maxwell's equation in the magnetic limit (Appendix A) are given by: Second equation implies that magnetic field is constant in space. Also, if magnetic field β 12 I is time independent (as we demand), electric field is curl free. But first equation already tells us that electric field should be divergencefree. Hence electric field is constant over the 2 dimensional space. However, electric field can still be time dependent, as the corresponding term does not appear in the magnetic limit of Maxwell's equations. Finally in our case we have, a constant electric field (may depend on time) and a constant magnetic field.
For this system, the form of j i S is given by  Tables (III).

VI. DISCUSSION
We started with a generic (d + 2)-dim non-conformal relativistic fluid (d ≥ 2) with anomalies specific to d = 2 in presence of background electromagnetic fields on flat space. Light cone reduction of this system gives a (d+1)dim non-relativistic fluid. We demand that our nonrelativistic fluid satisfies the second law of thermodynamics which imposes certain constraints on the system. For example we find that the parity odd terms in non-relativistic theory can only sustain for incompressible fluid in electromagnetic background with constant magnetic field in (2 + 1)-dim.
Our non-relativistic fluid (obtained by LCR) has generic dissipative terms, which are allowed by the symmetry and the condition that local entropy production is always positive definite. LCR does not constraint the size Eqn. (A.2). We will see that the magnetic limit essentially does the same. However there exists a consistent electric limit as well, where Bianchi identities are broken, but inhomogeneous Maxwell's equations are preserved. In terms of dimensionless fields and sources one can write the inhomogeneous Maxwell's equations as: and the Bianchi identities as: where α and β are some dimensionless constants given by: From here we can easily read out, that in non-relativistic limit β ≪ 0. To preserve the Bianchi identities therefore, E i should be one order smaller than F ij ; which is why this limit is called 'magnetic limit'. If we measure smallness in terms of a parameter ǫ ∼ 1/c, we will have E i ∼ ǫ n+1 and F ij ∼ ǫ n . Therefore from Eqn. (A.3), first nontrivial order of ρ ∼ αǫ n+1 , and J i ∼ αǫ n−1 , and inhomogeneous Maxwell's Equations (in conventional units) reduce to: Essentially we have just dropped the displacement current term from the Ampere's Law. To explicitly see if this limit is Galilean invariant, one can lookup [18]. These are the very equations that have been used in Section (IV A) and (V B).
One can now push the expansion of Eqn. (A.3) to ǫ n+2 order and derive the continuity equation: It should be remembered however that there are mixed orders in this equation and to the highest order it just states: ∇ i J i = 0, which also follows from Eqn. (A.7). Continuity equation takes its usual form only when first two leading orders of J i vanish and ρ ∼ J i , but in which case one of the Maxwell's equations modifies to: ∇ j F ji = 0. This in conjugation with the Bianchi identity would mean that the magnetic field is constant over space. Finally in conventional units this limit can be summarized as: F ij ∼ E i ∼ c −n , and to the maximum order ρ ∼ c −n ǫ o , J i ∼ c −n+2 ǫ o .