Thermostats and barostats Flashcards
Explain what a micro-canonical ensemble means.
The total energy in the system is constant. It is characterized by the particle dens.:
rho(r,p) = delta(H(r,p) - E_tot)
Equilibrium thermodyn. quantities can calc. as ensemble averages.
If taking time average if long time can hope it includes all configurations but not exactly.
How does temp. fluctuations depend on N?
sigma^2(T)/ = 2/(3N)
the variance of the temp. ie. the fluctuations are prop. to 1/sqrt(N)
Explain an canonical ensemble
The system is in thermodynamical equilibrium with a heat bath. All states are sampled according to their Boltzmann factor and don’t need to reweight the samples according to this. An ensemble with a given number of particles and given volume in thermal equilibrium with heat bath.
rho(r,p) = exp( -beta*H(r,p))
Explain the idea of the stochastic Andersen thermostat.
The Andersen thermostat stochastically resets the velocity of particles according to Maxwell-Boltzmann distribution.
The time in between collisions are Poisson distributed and in between the collisions the particles follow the Newtons eq. The particles to collide with are chosen according to the Boltzmann distribution.
This yields canonical ensemble since the particles are chosen according to Boltzmann, but the kinetics cannot be calc. from this since one changes the velocities of the particles.
Describe the Berendsen weak coupling scheme.
At some time interval tao one rescales the momenta to keep the temp. around the desired temp. T_0.
The rescaling goes like
v_new = v_old*(1+dt/tao (T_0/T-1))^1/2
tao is the relaxation time, the time in between two rescalings.
Berendsen solves the diff. eq.
dT/dt = 1/tao(T_0-T) –> T = T_0 - Cexp(-T/tao)
the temp. is changed exponentially toward the desired temp. T_0.
This gives only week pertubation of the system but yields not a canonical ensemble.
When tao = dt, the velocity is scaled at every time step and no fluctuations are allowed, on other hand when tao –> infinity Newtonian motion is recovered, NVE ensemble.
Explain briefly the extended system thermostat.
An artificial dynamic variable is added to the Lagrangian to make heat flow in/out of the real system, but the total energy of the extended system is kept constant.
L_nose = L + Q/2s’^2 - g/betaln(s)
But want ot have in real sys. when simulating so transform, micro-canonical ensemble in the extended sys. gives canonical ensemble for the real sys.
In Nose-hover the thermodyn. friction coeff. chi = s*p_s/Q decides how the heat flows.
chi 0 heat flows out of the real system.
What are the pros and cons of extended sys. thermostat?
+ it does not heavily disturb trajectories like in stochastic coupling
+ deterministic and time reversible
+ canonical ensemble, relaxation regulated by Q
+ same scheme can be used to regulated eg. the pressure
- more complex programming
- temp. oscillations can occur
if bad Q is chosen, this affects the friction coeff.
small Q gives high freq. oscillations, but if Q large chi –> 0 and have less coupling
Give two examples of problems connected to thermostatting.
The solvent is heated faster then the solute, since there are so many solvent molecules it is possible to have the solvent at 305K and the solute at only 200K and still get an average at 300K.
–> use different thermostats for solvent and solute.
Another problem may be that the kinetic energy accumulates and makes the whole sys. flies through space.
–> remove center-of-mass motion regularly or apply rototranslational constraints
Explain Berendsen barostat and how one can avoid temp. fluctuations due to pot. energy changes.
Berendsen barostat is applied by changing the coordinates and scale the box, ie. when changing the volume the pressure in the box also changes.
V_new = V_old(1+Kdt/tao(P-P_0))^1/3
this may cause significant changes in the pot. energy, conversion of pot. energy to kin. energy on short time scales may induce high freq. fluctuations in temp.
–> use smaller tao for thermostatting then for barostatting
