Gurzadyan-Savvidy relaxation

Using the geometric methods of theory of dynamical systems,[9][10][11] Gurzadyan and Savvidy showed the exponential instability (chaos) of spherical N-body systems interacting by Newtonian gravity and derived the collective (N-body) relaxation time (see also [12])

${\displaystyle \tau _{GS}=\left({\frac {15}{4}}\right)^{2/3}\left({\frac {1}{2\pi {\sqrt {2}}}}{\frac {v}{GMn^{2/3}}}\right),}$

where ${\displaystyle v}$ denotes the average stellar velocity, ${\displaystyle M}$ is the mean stellar mass and ${\displaystyle n}$ is the stellar density. Normalized for parameters of stellar systems like globular clusters it yields

${\displaystyle \tau _{GS}\simeq 10^{8}{\text{year}}\left({\frac {v}{10{\text{km/sec}}}}\right)\left({\frac {n}{1{\text{pc}}^{-3}}}\right)^{-2/3}\left({\frac {M}{M_{\odot }}}\right)^{-1}.}$

For clusters of galaxies it yields 10-1000 Gyr. Comparing this (GS) relaxation time to the 2-body relaxation time (see [13][14])

${\displaystyle \tau _{2b}={\frac {{\sqrt {2}}v^{3}}{\pi G^{2}M^{2}nln(N/2)}},}$

Gurzadyan and Savvidy obtain

${\displaystyle {\frac {\tau _{2b}}{\tau _{GS}}}\simeq {\frac {v^{2}}{GMn^{1/3}}}{\frac {1}{lnN}}\simeq {\frac {d}{r_{*}}}{\frac {1}{lnN}},}$

where ${\displaystyle r_{*}=GM/v^{2}}$ is the radius of gravitational influence and d is the mean distance between stars. With increasing density, d decreases and approaches ${\displaystyle r_{*}}$ so that the 2-body encounters become the dominant in the relaxation mechanism. The times ${\displaystyle \tau _{GS}}$ and ${\displaystyle \tau _{2b}}$ are related to the dynamical time ${\displaystyle \tau _{dyn}=D^{3/2}{GMN}^{1/2}}$ by the relations

${\displaystyle \tau _{GS}\simeq {\frac {D}{d}}\tau _{dyn},\quad \tau _{2b}\simeq {\frac {D}{r_{*}}}\tau _{dyn},}$

and reflect the fact of existence of 3 scales of time and length for stellar systems (see also [15][16][17][18])

${\displaystyle D:\tau _{dyn}\quad$ ;\quad d:\tau _{GS}\quad ;\quad r_{*}:\tau _{2b}.}

That approach (from the analysis of so-called two-dimensional curvature of the configuration space of the system) enabled to conclude[19] that while the spherical systems are exponentially instable systems (Kolmogorov K-systems), the spiral galaxies "spend a large amount of time in regions with positive two-dimensional curvature" and hence "elliptical and spiral galaxies should have a different origin". Within the same geometric approach Gurzadyan and Armen Kocharyan had introduced the Ricci curvature criterion for relative instability (chaos) of dynamical systems.[20][21][22]

GS-time scale ${\displaystyle \tau _{GS}=\tau _{dyn}N^{1/3}}$ has been rederived by Gurzadyan and Kocharyan using stochastic differential equation approach[23]

Observational support to the GS-time scale is reported for globular clusters.[24] Numerical simulations supporting GS-time scale are claimed in.[25][26][27][28]