Limiting behaviors of $\eta$ for the thermalized $\beta$-FPU chain

We change variables $y_j=q_j-q_{j+1}$ in the Hamiltonian (29) for $\beta$-FPU to obtain

$$H(p,y)=\sum_{j=1}^N\left[\frac{1}{2}p_j^2+\frac{1}{2}y_j^2+\frac{\beta}{4}y_j^4\right].$$ (74)

Next, we compute the pdf's for the momentum and displacement. Any $p_j$ is distributed with the Gaussian pdf $z_p=C_p\exp(-{\theta }^{-1}p^2/2)$ and any $y_j$ is distributed with the pdf $z_y=C_y\exp(-{\theta }^{-1}(y^2+\beta y^4/2)/2)$, where $C_p$ and $C_y$ are the normalizing constants. As we have discussed, the renormalization factor $\eta$ of the $\beta$-FPU system in thermal equilibrium is given by Eq. (25), and its approximation via the self-consistency argument $\eta _{sc}$ is given by Eq. (52). Here, we compare the behavior of both formulas in two limiting cases, i.e., the case of small nonlinearity $\beta\rightarrow 0$ and the case of strong nonlinearity $\beta\rightarrow\infty$. We will use the following expressions for the average density of kinetic, quadratic potential and quartic potential parts of the total energy of the system

$$\frac{\langle K\rangle }{N} = \frac{1}{N}\sum_{j=1}^N\frac{\langle p_j^2\rangle }{2}=\frac{1}{2}\theta ,$$ (75)

$$\frac{\langle U\rangle }{N} = \frac{1}{N}\sum_{j=1}^N\frac{\langle y_j^2\rangle }{2}=\frac{1}{2}\langle y^2\rangle ,$$ (76)

$$\frac{\langle V\rangle }{N} = \frac{\beta}{4N}\sum_{j=1}^N\langle y_j^4\rangle =\frac{\beta}{4}\langle y^4\rangle .$$ (77)

In a canonical ensemble, the temperature of a system is given by the temperature of the heat bath. By identifying the average energy density of the system with $\bar{e}=E/N$ in our simulation (a microcanonical ensemble), we can determine $\theta$ as a function of $\bar{e}$ and $\beta$ by the following equation

$$\frac{1}{N}\big(\langle K\rangle +\langle U\rangle +\langle V\rangle \big)=\bar{e}.$$ (78)

We start with the case of small nonlinearity $\beta\rightarrow 0$. Suppose in the first order of the small parameter $\beta$ the temperature has the following form

$$\theta (\beta)=\theta _0+\beta \theta _1,$$ (79)

where $\theta _0=O(1)$ and $\theta _1=O(1)$. We find the values of $\theta _0$ and $\theta _1$ using the constraint (A5). We use the following expansions in the small parameter $\beta$

$$\int_{-\infty}^{\infty}e^{-\frac{1}{2\theta (\beta)}(y^2+\beta\frac{y^4}{2})}~dy$$

$$=\sqrt\frac{\pi}{8}\sqrt{\theta _0}\left(4+\Big(\frac{2\theta _1}{\theta _0}-3\theta _0\Big)\beta\right)+O(\beta^2),$$ (80)

$$\int_{-\infty}^{\infty}y^2e^{-\frac{1}{2\theta (\beta)}(y^2+\beta\frac{y^4}{2})}~dy$$

$$=\sqrt\frac{\pi}{8}\sqrt{\theta _0}\left(4\theta _0+(6\theta _1-15\theta _0^2)\beta\right)+O(\beta^2),$$ (81)

$$\int_{-\infty}^{\infty}(y^2+\frac{\beta}{2}y^4)e^{-\frac{1}{2\theta (\beta)}(y^2+\beta\frac{y^4}{2})}~dy$$

$$=\sqrt\frac{\pi}{8}\sqrt{\theta _0}\left(4\theta _0+(6\theta _1-9\theta _0^2)\beta\right)+O(\beta^2).$$ (82)

Then, in the first order in $\beta$, Eq. (A5) becomes

$$\theta _0+\beta \theta _1+\frac{\sqrt\frac{\pi}{8}\sqrt{\theta _0... ...eft(4+ \Big(\frac{2\theta _1}{\theta _0}-3\theta _0\Big)\beta\right)}=2\bar{e},$$

and we obtain $\theta _0=\bar{e}$ and $\theta _1=(3/4)\bar{e}^2$. Therefore, for the average kinetic energy density, we have

$$\frac{\langle K\rangle }{N}=\frac{1}{2}\bar{e}+\frac{3}{8}\bar{e}^2\beta+O(\beta^2),$$ (83)

and, for the average quadratic potential energy density, we have

$$\frac{\langle U\rangle }{N} = \frac{1}{2}\frac{\sqrt\frac{\pi}{8}\sqrt{\theta _0}\left(4\theta ... ...heta _0} \left(4+\Big(\frac{2\theta _1}{\theta _0}-3\theta _0\Big)\beta\right)}$$

$$= \frac{1}{2}\bar{e}-\frac{9}{8}\bar{e}^2\beta+O(\beta^2).$$ (84)

Finally, we obtain Eq. (53), i.e., for small $\beta$

$$\eta=1+\frac{3}{2}\bar{e}\beta+O(\beta^2).$$ (85)

Similarly, from Eq. (52), we find the small $\beta$ limit of the approximation $\eta _{sc}$

$$\eta_{sc}=1+\frac{3}{2}\bar{e}\beta+O(\beta^2).$$

Now, we consider the case of strong nonlinearity $\beta\rightarrow\infty$. From Eq. (A5), we conclude that temperature in the large $\beta$ limit, which we denote as $\theta_{\infty}$, stays bounded, i.e., $0<\theta_{\infty}<2\bar{e},$ and, in the limit of large $\beta$, we obtain for Eq. (A5)

$$\theta_{\infty}+\frac{\int_{-\infty}^{\infty}\frac{\beta}{2}y^4e^... ...dy}{\int_{-\infty}^{\infty}e^{-\frac{\beta}{4\theta_{\infty}}y^4}~dy}=2\bar{e}.$$ (86)

After performing the integration, we obtain $\theta_{\infty}=(4/3)\bar{e},$ and the average kinetic energy density becomes $\langle K\rangle /N=(2/3)\bar{e}$. For the average quadratic potential energy density, we have

$$\frac{\langle U\rangle }{N}=\frac{1}{2}\frac{\int_{-\infty}^{\inf... ...{3}{4})}{\Gamma(\frac{1}{4})}\left(\frac{4\bar{e}}{3\beta}\right)^{\frac{1}{2}}$$ (87)

For the renormalization factor, we obtain the following large $\beta$ scaling

$$\eta=\sqrt{\frac{\Gamma(\frac{3}{4})}{\sqrt{3}\Gamma(\frac{1}{4})}}\bar{e}^\frac{1}{4}\beta^\frac{1}{4}.$$ (88)

Similarly, for the approximation of $\eta _{sc}$, we obtain $A=C\sqrt{\bar{e}\beta}$, $B=4\bar{e}\beta$, and $C=2\sqrt{3}\Gamma(3/4)/\Gamma(1/4)$. Therefore, the large $\beta$ scaling of $\eta _{sc}$ becomes

$$\eta_{sc}=\sqrt{\frac{C+\sqrt{C^2+16}}{2}}\bar{e}^\frac{1}{4}\beta^\frac{1}{4},$$ (89)

which yields Eq. (54).