Newtonian Kinetic Theory and the Ergodic-Nonergodic Transition

Abstract

In a recent work we have discussed how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. In the present work we use this development for investigating ergodic-nonergodic (ENE) transitions in dense fluids. The theory is developed in terms of a core problem spanned by the variables ρ, the number density, and B, a response density. We set up the perturbation theory expansion for studying the self-consistent model which gives rise to a ENE transition. Our main result is that the low-frequency dynamics near the ENE transition is the same for Smoluchowski and Newtonian dynamics. This is true despite the fact that term by term in a density expansion the results for the two dynamics are fundamentally different.

Appendices

Appendix A: Three-point Vertex Functions

We obtain the FDT relations between the different three-point vertex functions here. For this we use the general definition of the three point correlation functions and the corresponding three-point vertex function as,

(A.1)

where α _i stands for the field α at the point i and so on where everything is in Fourier space. The labels α, μ and ν are respectively taken from the set {ρ,B}. Using the result that $\varGamma_{\rho_{i}\rho_{j}\rho_{k}}=0$, we obtain from the formula (A.1), the various cumulants are:

(A.2)

(A.3)

(A.4)

Substituting the results (A.2)–(A.4) in Eq. (3.54), we obtain an expansion in terms of the various three-point vertex functions. For example the coefficient of the vertex function $\varGamma_{B_{i}\rho_{j}\rho_{k}}$ is obtained after some trivial but tedious algebra as

(A.5)

In reaching the above result we have used the FDT relation (3.48). Similarly for the other two vertices with one B field, i.e., $\varGamma_{\rho_{i}B_{j}\rho_{k}}$ and $\varGamma_{\rho_{i}\rho_{j}B_{k}}$ are obtained. For the vertex with two B fields, e.g, $\varGamma_{B_{i}B_{j}\rho_{k}}$ we obtain the corresponding coefficient as

(A.6)

and similarly we obtain the coefficients of the other two vertices each with two B fields. After organizing the coefficients of the different three-point vertex functions the following result is obtained.

(A.7)

In reaching the above result we have dropped a nonzero common factor of G _ρB (i)G _ρB (j)G _ρB (k) from the LHS.

Next, consider the relations (3.55) involving three-point cumulants having two B fields. Once again substituting the results (A.3)–(A.4) in Eq. (3.55), we obtain after organizing the coefficients of the different three-point vertex functions the following result.

(A.8)

In obtaining the above result, we have dropped the common factor of G _ρB (i)G _ρB (j) from both sides and used the basic FD relation (3.48). Using Eq. (3.65) in the result (A.8) we find that the coefficient of G _ρB (k) vanishes and one has the result

(A.9)

Appendix B: Higher-Order Thermodynamic Sum Rule

2.1 B.1 Three-point Quantities

Let us consider the low frequency behavior of the full three-point cumulant G _ρBB (123). The quantity ρ(1) in G _ρBB (123) can be replaced by an arbitrary function of density as long as each density corresponds to the same time. The three-point vertex Γ _Bρρ (123) is related to the 3-point correlation G _ρBB by the general relation

(B.1)

Among the FDR identities for the 3-point cumulants, we have for the imaginary part of G _ρBB the relation:

(B.2)

Next, look at the definition of the inverse time Fourier transform

(B.3)

where we have introduced

(B.4)

In this notation we do not always write the third frequency entry since it is implied:

(B.5)

and the wave number dependence has been suppressed. Setting t ₂=t ₃ in Eq. (B.3) we obtain,

(B.6)

This vanishes for t ₂>t ₁. This is consistent with

(B.7)

being analytic in the UHP for ω ₁. Assuming $\widetilde{G}_{\rho BB}(\omega_{1},\omega_{2})$ is analytic in the UHP for ω ₁ we can write a dispersion relation

(B.8)

Putting Eq. (B.4) in Eq. (B.2) gives

(B.9)

Putting Eq. (B.9) in Eq. (B.8) gives

(B.10)

Letting ω ₁ and ω ₂ go to zero gives

(B.11)

We have a FDR identity

(B.12)

which tells us that

(B.13)

and we can write

(B.14)

In the time domain

(B.15)

Fourier transforming over t ₂ obtains

(B.16)

Setting t ₃=t ₁ and letting ω ₂→0:

(B.17)

Combining Eqs. (B.14) and (B.17) gives

(B.18)

When the times of the ρ’s are equal in G _ρBρ we have

(B.19)

Introducing f(t ₁)=δρ(q ₁ t ₁)δρ(q ₂,t ₁) one has G _fB (0) satisfies the two-time FDR

(B.20)

(B.21)

and the static three-point cumulant enters the development. We have then

(B.22)

In terms of the three-point vertex

(B.23)

which is a result of much use in evaluating the one-loop contribution to the self energy both at single-particle and collective levels. Note that γ _ρρρ is a static three-point vertex.

Appendix C: FDR Matrix Propagators

FDR matrix propagators (FDRMP) A _μν (q,ω) satisfy the following properties:

(C.1)

From this it follows

(C.2)

Finally the element A _BB (q,ω)=0.

We now prove the following important property of the FDRMP: If A _αβ (q,ω) and C _αβ (q,ω) are FDR matrix propagators then

(C.3)

is also a FDR matrix propagator. The proof is rather direct. Look first at the response channel:

(C.4)

Consider next the off-diagonal components

(C.5)

It is easy to see that $D_{\rho B}(q,\omega)= D^{*}_{B\rho }(q,\omega)$. Next consider the diagonal component

(C.6)

The above result implies that

(C.7)

(C.8)

Together Eqs. (C.7) and (C.8) give $D_{\rho\rho}(q,\omega ) = D^{*}_{\rho\rho}(q,\omega )$.

Let us now consider the dressed propagators respectively denoted as $\bar{G}$ and $\widetilde{G}$. In operator notation $\bar{G}$ and $\widetilde{G}$ are respectively defined as $\bar{G} = \mathcal{G}\sigma{G}$ and $\widetilde{G}= \mathcal{G}\sigma G \sigma \mathcal{G}$. Writing out explicitly the matrix forms we obtain for $\bar{G}$ and $\widetilde{G}$ the following expressions:

(C.9)

(C.10)

From the above theorem then it follows that both $\bar{G}$ and $\widetilde{G}$ are FDRMP. This holds since each of the G, $\mathcal{G}$, and G ⁽⁰⁾ satisfies the conditions of being a FDRMP.