The surface integral of the function $f$ over the surface $S$ is denoted by
\[ \int\int_S f dS \]
$dS$ is the area of an infinitesimal piece of the surface $S$.
Average over the sphere is
\[ <f> = \frac{1}{4\pi} \int_0^{2\pi} d\phi \int_0^{\pi} f \sin(\theta)d\theta ,\]
where $dS=\sin(\theta)d\theta d\phi$. This can be seen as a weight average, the weight is surface area.
\[ <\cos^2(\theta)>=1/3 \]
\[<\sin(\theta)>=2/3.\]
From: https://dynesty.readthedocs.io/en/stable/examples.html.Slurm (Simple Linux Utility for Resource Management) is a group utilities used for managing workloads on computer clusters. It can also be sued in your personal laptop.
$sudo dnf install slurm slurm-slurmd slurm-slumrctld $sudo -u munge /usr/sbin/mungekey -v $sudo systemctl start munge $sudo systemctl status munge $sudo systemctl enable munge $munge -n | unmunge // check munge works $hostname -f // find out hostname $sudo vi /etc/slurm/slurm.conf
ClusterName=localcluster
SlurmctldHost=localhost // or hostname
MpiDefault=pmix
ProctrackType=proctrack/cgroup
ReturnToService=1
SlurmctldPidFile=/var/run/slurm/slurmctld.pid
SlurmctldPort=6817
SlurmdPidFile=/var/run/slurm/slurmd.pid
SlurmdPort=6818
SlurmdSpoolDir=/var/spool/slurm/d
SlurmUser=root
StateSaveLocation=/var/spool/slurm/ctld
SwitchType=switch/none
InactiveLimit=0
KillWait=30
MinJobAge=300
SlurmctldTimeout=120
SlurmdTimeout=300
Waittime=0
Reference:
https://src.fedoraproject.org/rpms/munge
http://docs.nanomatch.de/technical/SimStackRequirements/SingleNodeSlurm.html
https://blog.llandsmeer.com/tech/2020/03/02/slurm-single-instance.html
The kinetic energy between two colliding particles can be expressed in terms of center of mass and relative velocity by
\[ E_k = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2v_2^2 = \frac{1}{2} M v_c^2 + \frac{1}{2} \mu v_r^2 , \]
a sum of a center-of-mass term and a relative momentum term.
where $M$, $v_c$, $\nu$, and $v_r$ are given by
\[ M = (m_1+m_2) ; \]
\[ \mathbf{v_c} = \frac{m_1 \mathbf{v_1} + m_2 \mathbf{v_2}}{m_1 +m_2} ;\]
\[ \mu = \frac{m_1 m_2}{m_1 + m_2} ;\]
\[ \mathbf{v_r} = \mathbf{v_1} - \mathbf{v_2} .\]
$\frac{1}{2} \mu v_r^2$ is called collision energy.
amu is short for atomic mass unit. Protons have a positive electrical charge of one (+1) and a mass of 1 atomic mass unit (amu), which is about 1.67×10−27 kilograms.
Reference:
https://web.chem.ox.ac.uk/teaching/Physics%20for%20CHemists/Mechanics/Collisions.html
https://www.angelo.edu/faculty/kboudrea/periodic/structure_mass.htm
