Quantum-friction.md

 # Introduction
-Quantum friction is an external potential added to the Hamiltonian that breaks time-reversal invariance so as to cool the system (decrease its total energy). It may be used to cool fermionic many-body systems with thousands of wavefunctions that must remain orthogonal. It is described in details in:  
+Quantum friction is an external potential added to the Hamiltonian that breaks time-reversal invariance so as to cool the system (decrease its total energy). It may be used to cool fermionic many-body systems with thousands of wavefunctions that must remain orthogonal. It is described in detail in:  
+* J. E. Alba-Arroyo, D, Pęcak, M, M, Forbes, G, Wlazłowski, _Local Quantum Friction with Pairing: Unitary Dissipation in Large Fermi Systems_, [arXiv:2512.12866](https://arxiv.org/abs/2512.12866) 
 * A. Bulgac, M. M. Forbes, K. J. Roche, G. Wlazłowski,
 _Quantum Friction: Cooling Quantum Systems with Unitary Time Evolution_,
 [arXiv:1305.6891](https://arxiv.org/abs/1305.6891) 
 
-The quantum friction potential is given by:
-```math
-  V_{\sigma}^{(qf)} = -\alpha \frac{\hbar\,\vec{\nabla}\cdot\vec{j}_\sigma}{\rho_0}
-```
-where $`\rho_0=\frac{k_F^3}{6\pi^2}`$ is reference density. By construction, this potential removes any irrotational currents. Thus it provides a convenient method of removing phonon excitations from the system.
-
 # Usage
-The quantum friction is controlled via `input` file via tags:
+The quantum friction is controlled via the `input` file via tags:
 ```bash
 # --------------- QUANTUM FRICTION ------------------
-# qfalpha                 0.0     # alpha parameter for quantum friction term
-# qfstart                 0.0     # start time for evolving with quantum friction [eF]
-# qfstop                  0.0     # stop time for evolving with quantum friction [eF]
-# qfswitch                0.0     # the friction will be activated and deactivated gradually over this period of time [eF] 
+# See: Wiki -> Quantum friction
+# qfalpha                 0.0     # alpha parameter for quantum friction term (cooling in the normal channel), default=0
+# qfbeta                  0.0     # beta parameter for quantum friction term (cooling in the pairing channel), default=0
+# qfgamma                 0.0     # particle control parameter (cooling in the pairing channel), default=0
+# qfNreq                  0.0     # requested particle number, meaningful only if qfgamma>0
+# qfstart                 0.0     # start time for evolving with quantum friction [eF], default=0
+# qfstop                  0.0     # stop time for evolving with quantum friction [eF], default=infinity
+# qfswitch                0.0     # the friction will be activated and deactivated gradually over this period of time [eF], default=1
 ```
+For the meaning of these variables, see [arXiv:2512.12866](https://arxiv.org/abs/2512.12866)
 *Notes*:
-* `qfalpha ~ 1` looks to be reasonable choice,
-* too large value of `qfalpha` may lead to instability of the code, typically it manifests via growing of the energy during the evolution.  
+* `qfalpha ~ 5-10` provides optimal damping efficiency.
+* `qfbeta ~ 5` provides optimal damping efficiency.
+* `qfgamma ~ 0.1` provides optimal damping efficiency.
+* A too large value of `qfalpha` may lead to instability of the code; typically, it manifests via the growth of the energy during the evolution.  
+* 
 
 # Example
-Consider application a time-dependent potential:
+Consider the application a time-dependent potential:
 ```math
 V_{\textrm{ext}}(x,y,t)=s(t,t_{\textrm{start}}, t_{\textrm{stop}})\exp\left[-\frac{x^2}{2\sigma_x^2}-\frac{y^2}{2\sigma_y^2}\right]
 ```
- to the unitary Fermi gas, being initially in the uniform state, where $`s(t,t_{\textrm{start}}, t_{\textrm{stop}})`$ is (smooth) step function that acquires 1 in time interval $`[t_{\textrm{start}}, t_{\textrm{stop}}]`$, otherwise is 0. Implemamntion of this potential is following (`problem-definition.h`):
+ to the unitary Fermi gas, being initially in the uniform state, where $`s(t,t_{\textrm{start}}, t_{\textrm{stop}})`$ is (smooth) step function that acquires 1 in time interval $`[t_{\textrm{start}}, t_{\textrm{stop}}]`$, otherwise is 0. Implementation of this potential is following (`problem-definition.h`):
 ```c
 __device__ __host__ inline double switch_function(double t, double T, double alpha)
 {
@@ -91,7 +94,7 @@ When executing this code for lattice (`predefines.h`):
 #define NZ 8
 #define FUNCTIONAL ASLDA
 ```
-with input file parmaters:
+with input file parameters:
 ```bash
 params0     0.25 # gauss amplitude [eF]
 params1     2.0 # width in x direction
@@ -106,7 +109,7 @@ qfstop                  250.0
 qfswitch                1.0
 # ...
 ```
-the resulting evolution of the total energy looks like this:
+The resulting evolution of the total energy looks like this:
 ![qf](uploads/0a927351fb183a211fb554689fdd30cb/qf.png)
 
-It is observed, that time-dependent potential excites the system. Selected potential does not introduce angular momentum to the system, and only phonons are induced. When evolving this state with the quantum friction (`qfalpha=1.0`) we see that the system returns after some time to its ground state. The origin for energy fluctuations for `qfalpha=0.0` and for $`t\varepsilon_F>50`$ see [here](Regularization schemes of the pairing field).
+It is observed that the time-dependent potential excites the system. Selected potential does not introduce angular momentum to the system, and only phonons are induced. When evolving this state with quantum friction (`qfalpha=1.0`), the system returns to its ground state after some time. The origin of energy fluctuations for `qfalpha=0.0` and for $`t\varepsilon_F>50`$ see [here](Regularization-schemes-of-the-pairing-field).
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