Long-time Analysis of statistical behavior

The analysis proceeds by first forming the hierarchy of equations for the spectral cumulants (correlation functions of the wave amplitudes) defined as follows. The mean is zero.

$$\left<a^s({\bf k})a^{s'}({\bf k}')\right> =\delta({\bf k}+{\bf k}')q^{s s'}({\bf k}, {\bf k}'),$$ (52)

$$\left<a^s({\bf k})a^{s'}(k')a^{s''}({\bf k}'')\right>$$

$$= \delta({\bf k}+{\bf k}'+{\bf k}'') q^{s s' s''} ({\bf k},{\bf k}',{\bf k}''),$$ (53)

$$\left<a^{s}({\bf k})a^{s'}({\bf k}') a^{s''}({\bf k}'')a^{s'''}({\bf k}''')\right>$$

$$= \delta({\bf k}+{\bf k}'+{\bf k}''+{\bf k}''' )q^{s s' s'' s'''}({\bf k},{\bf k}',{\bf k}'',{\bf k}''')$$

$$+\delta({\bf k}+{\bf k}')\delta({\bf k}''+{\bf k}''') q^{s s'} ({\bf k},{\bf k}') q^{s''s'''}({\bf k}'',{\bf k}''')$$

$$+\delta({\bf k}+{\bf k}'')\delta({\bf k}'+{\bf k}''') q^{s s''} ({\bf k},{\bf k}'') q^{s's'''} ({\bf k}',{\bf k}''')$$

$$+ \delta({\bf k}+{\bf k}''')\delta({\bf k}'+{\bf k}'') q^{s s'''} ({\bf k},{\bf k}''') q^{s' s''}({\bf k}',{\bf k}''),$$ (54)

where $\left<\dots\right>$ denotes average and the presence of the delta function is a direct reflection of spatial homogeneity. Indeed the property of spatial homogeneity affords one a way of defining averages, which does not depend on the presence of a joint distribution. We can define the average $\left<\eta({\bf x}) \eta({\bf x}+{\bf r})\right>$ as simply an average over the base coordinate, namely

$$\left<\eta({\bf x}) \eta({\bf x}+{\bf r})\right>=\frac{1}{(2... ...} \int_{-L}^L\eta({\bf x})\eta({\bf x}+{\bf r}) d {\bf x}\ .$$ (55)

To derive the main results of this paper, it is sufficient to write the equations for the second and third order cumulants. They are

$$\frac{d q^{s s'}_{{\bf k}{\bf k}'}}{d t}=\epsilon P_{00'} \sum_{s... ... k},{\bf k}_p,{\bf k}_q}^{s,s_p,s_q} q_{{\bf k}'{\bf k}_p {\bf k}_q}^{s's_ps_q}$$

$$\times \exp [i(s_p\omega_p+s_q\omega_q-s\omega_k)t] \delta({{\bf k}-{\bf p}-{\bf q}}),$$

$${\bf k}+{\bf k}'=0\ ;$$ (56)

$$\frac{d q^{s s's''}_{{\bf k}{\bf k}'{\bf k}''}}{d t}=\epsilon P_{... ... k}'', {\bf k}_p, {\bf k}_q}^{s',s'',s_p,s_q} \delta({{\bf k}-{\bf p}-{\bf q}})$$

$$\times \exp [i(s_p\omega_p+s_q\omega_g-s\omega_k)t]+$$

$$2 \epsilon P_{00'0''}\sum_{s_ps_q}L_{{\bf k},-{\bf k}',-{\bf k}''... ...s,s_p,s_q} q^{s',s_p} _{{\bf k}', -{\bf k}'} q^{s'',s_q}_{{\bf k}'',-{\bf k}''}$$

$$\times \exp [i(s_p\omega'+s_q\omega''-s\omega)t]\,,\qquad {\bf k}+{\bf k}'+{\bf k}''=0$$ (57)

where the symbol $P_{00'}$ ($P_{0 0' 0''}$) means that we sum over all replacements $0\rightarrow 0', \ \ 0'\rightarrow 0$ ( $0\rightarrow 0',0'\rightarrow 0'',\ 0''\rightarrow 0,\ 0\rightarrow 0'', 0'\rightarrow 0,\ \ 0''\rightarrow 0'$).

The total energy of the system per unite volume can be written as

$$\lim_{r\rightarrow 0} \langle \frac{1}{2}\rho_0 v_j({\bf x}) v_j({\bf x}+{\bf r})$$

$$+ \frac{c^2\rho_0}{\mu}\eta({\bf x}) \eta({\bf x}+{\bf r}) +\frac{\rho c^2}{2\mu} (\mu-2)\eta({\bf x})\eta({\bf x} + {\bf r})\rangle$$

$$= \lim_{{\bf r}\rightarrow 0}\sum _{s_1 s_2} \int \frac{\rho_0c^2 \epsilon ^2}{2}(1-s_1s_2)q^{s_1s_2} ({\bf k})e^{i{\bf k} {\bf r}}d {\bf k}$$

$$= \lim_{{\bf r}\rightarrow 0}\int\rho_0c^2 \epsilon ^2(q^{+-}({\bf k})+q^{-+} ({\bf k}))e^{i{\bf k}{\bf r}} d {\bf k}$$

$$= \int 2\rho_0c^2 \epsilon ^2q^{+-}({\bf k})d {\bf k}$$ (58)

since $q^{+-}({\bf k})=q^{-+}(-{\bf k})$. The spectral energy is therefore $2\rho c^2 \epsilon ^2 q^{+-}({\bf k})$. For convenience we denote $q^{+-}({\bf k})$ as $e({\bf k})$.

To leading order in $\epsilon$, $q^{s s'} ({\bf k}, {\bf k}')$ and $q^{s s' s''} ({\bf k}, {\bf k}',{\bf k}'')$ (which we may call $q^{s s'}_0 ({\bf k}, {\bf k}')$ and $q^{s s' s''}_0 ({\bf k}, {\bf k}',{\bf k}'')$) are independent of time. Anticipating, however that certain parts of the higher order iterates in their asymptotic expansions may become unbounded, we will allow both $q^{s s'}_0 ({\bf k}, {\bf k}')$ and $q^{s s' s''}_0 ({\bf k}, {\bf k}',{\bf k}'')$ to be slowly varying in time

$$\frac{d q_0^{s s'}({\bf k},{\bf k}')}{d t} = \epsilon ^2 F_2^{s s'}\,,$$

$$\frac{d q_0^{s s' s''}({\bf k},{\bf k}',{\bf k}'')}{d t} = \epsilon ^2 F_3^{s s' s''}$$ (59)

and we will choose $F_2$ and $F_3$ to remove those terms with unbounded growth from the later iteration. We will find that for $s'=-s$, $F_2^{s -s}$ is given by the right-hand side of acoustic KE:

$$F_2^{s s'}= q_0^{s\ s'}({\bf k},{\bf k'}) \lim_{ \epsilon ^2\righ... ... s_p}{s \omega}\int\left(L_{ {\bf k},{\bf k}_p,{\bf k}_q} ^{s,s_p,s_q}\right)^2$$

$$\times q^{s_p,-s_p}({\bf k}_p)\Delta(s_p\omega_p+s_q\omega_q-s\omega) \delta({\bf k}_p+{\bf k}_q-{\bf k})d {\bf k}_p d {\bf k}_q$$

$$+\lim_{ \epsilon ^2\rightarrow 0 }\sum_{s_p s_q} \int \frac{s_q s_p}{s \omega}\int\left(L_{ {\bf k},{\bf k}_p,{\bf k}_q} ^{s,s_p,s_q}\right)^2$$

$$\times q_0^{s\ s'}({\bf k_q},{-\bf k_q})q^{s_p,-s_p}({\bf k}_p) \Delta(s_p\omega_p+s_q\omega_q-s\omega)$$

$$\delta({\bf k}_p+{\bf k}_q-{\bf k})d {\bf k}_p d {\bf k}_q$$

and that $F_2^{s s}$ and $F_3^{s s' s''}$ have the form

$$i q_0^{s s}({\bf k}, {\bf k}')(\bar \Omega_{\bf k}^s + \bar\Omega_{{\bf k}'}^s)$$ (60)

and

$$i q_0^{s s' s''}({\bf k}, {\bf k}',{\bf k}'')(\bar \Omega_{\bf k}^s + \bar\Omega_{{\bf k}'}^{s'} + \bar\Omega_{{\bf k}''}^{s''} )$$ (61)

respectively. It is clear that $\bar\Omega^s_k$ can be interpreted as a complex frequency modification. Its exact expression is given by

$$\bar\Omega_{\bf k}^s= - 4 i \lim_{ \epsilon ^2\rightarrow 0 }\sum... ... s_p}{s \omega}\int\left(L_{ {\bf k},{\bf k}_p,{\bf k}_q} ^{s,s_p,s_q}\right)^2$$ (62)

$$\times q^{s_p,-s_p}({\bf k}_p)\Delta(s_p\omega_p+s_q\omega_q-s\omega) \delta({\bf k}_p+{\bf k}_q-{\bf k})d {\bf k}_p d {\bf k}_q$$

and, when calculated out, is precisely equal to $s(\omega-c\vert{\bf k}\vert)\epsilon ^2$ in (1.17). Note that in (3.11), $t=T\epsilon ^2$ and $T$ is finite. The $\ln(1/ \epsilon ^2)$ coefficient comes from the term $\ln t$ or $\ln (T/ \epsilon ^2)= \ln T + \ln (1/\epsilon ^2)$ in the asymptotic expansion. For finite $T$, the dominant part is $\ln(1/ \epsilon ^2)$.

The perturbations method has the advantage that it is relatively simple to execute. However, there is no a priori guarantee that terms appearing later in the formal series cannot have time dependencies which mean they affect the leading approximations on time scales comparable to or less than $\epsilon ^{-2}$ (e.g. a term $\epsilon^4t^3$ should be accounted for before the term $\epsilon^2t$). To check this, one must have a systematic approach for exploring all orders in the formal perturbation series and removing (renormalizing) in groups those resonances which make their cumulative effects at time scales $\epsilon^{-N}(\ln(\frac{1}{\epsilon})^{-M})$, $N,M=1,2,3, ...$. The diagram approach, which requires some familiarity to execute, is designed to do this and, both for completeness and the fact that we will have to proceed beyond the one-loop approximation to resolve the questions of the angular redistribution of spectral energy, we include it here.