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dissipation of energy in gases or liquids due to a moving body is at the core of physical theories such as hydrodynamics , statistical physics or kinetic theory . some interesting new results at a fundamental level , regarding the connection of equilibrium and non  equilibrium free energy differences , as the jarzynski equation , or the relation between dissipation and the asymmetry of time 
reversal processes are known@xcite .
these results have been extended from classical to quantum mechanical systems @xcite . in this work
we investigate the dissipation process in gases using the boltzmann equation ( be ) with a time 
dependent external potential .
the main motivation for considering this problem was the ambivalent behavior of elastic collisions during the potential deformation : on the one hand collisions do not change the energy of the colliding particles , and this is also true for a time 
dependent potential , since the collisions are described as pointlike in phase space . on the other hand
it is immediately clear that the energy of the gas in a time 
dependent potential must depend on the collisions during the potential change .
it makes a difference in the distribution function if in the be collisions are present or not . the first impulse to investigate such problems using kinetic theory was the study of bose 
einstein condensation in trapped gases , where one can reach the condensation of a bose gas by changing the trapping potential @xcite .
+ the paper is organized as follows : sec .
[ sect2 ] gives a short overview of the theoretical elements and formulas used to derive the main results represented in sec .
[ sect3 ] . sec .
[ sect4 ] gives a numerical example and sec .
[ sect5 ] conclusion and outlook .
the appendix contains a proof of the h 
theorem for time 
dependent potentials and the relaxation time ansatz in the be .
first we present some general equations of kinetic theory used in this paper .
the central quantity to describe the system is the time 
dependent one particle distribution function @xmath0 .
the distribution function depends on the one particle phase space coordinates @xmath1 ( momentum and space coordinates , sometimes suppressed to simplify notation ) and time @xmath2 ( also suppressed sometimes ) .
@xmath3 is normalized to the number of particles @xmath4 , @xmath5 here @xmath6 denotes the volume element in the 6 dimensional phase space ( we omit always planck s constant @xmath7 and the boltzmann constant @xmath8 ) .
we consider a gas in a time 
dependent external potential @xmath9 .
the time evolution of the distribution function is given by the boltzmann equation ( see for example @xcite ) @xmath10 here @xmath11 denotes the collision integral , which we approximate by the relaxation time ansatz , @xmath12 because the external potential is time 
dependent the equilibrium distribution @xmath13 becomes time  dependent as well , @xmath14\}~.\ ] ] here @xmath15 is the time 
dependent chemical potential , @xmath16 corresponds to the time 
dependent inverse temperature @xmath17 and @xmath18 is the one particle hamiltonian for particles with mass @xmath19 and without inner degrees of freedom .
the equilibrium distribution eq .   does not contain a macroscopic momentum or angular momentum because we restrict ourselves to spherical symmetric potentials to make the discussion transparent .
the main result eq .
is not affected by this restriction , as the derivations show , since the explicit structure of @xmath20 does not enter .
so let us assume that the number of particles and the energy for a fixed time are the only conserved quantities , which allow the determination of @xmath21 and @xmath15 in eq .  .
in particular we have the relation @xmath22 ensuring conservation of the number of particles .
the conservation of energy @xmath23 for a fixed time is expressed as @xmath24 eq .   with an initial condition ( in our case a maxwell  boltzmann distribution ) and the two conditions eq .   and eq .   are the basis for our analytical and numerical studies in sec .
[ sect4 ] .
we remark that eq .   due to
eq .   and
eq .   becomes non linear , in contrast to the focker 
planck equation , which is a linear evolution equation .
a central quantity is the nonequilibrium entropy of the gas . for a given one particle distribution function
it is computed as @xcite @xmath25\mathrm{d}\gamma~.\ ] ] one can proof the validity of the h  theorem , @xmath26 with the relaxation time ansatz and a time 
dependent external potential also .
for details see the appendix app .
[ app1 ] . here
we need an additional expression for @xmath27 for gases not far from equilibrium .
we split the distribution function as @xmath28 , where @xmath29 denotes the small deviation of @xmath3 from the equilibrium .
expansion of the logarithm in eq .
up to second order and some lines of algebraic simplifications yields @xmath30 here @xmath31 is the time 
dependent equilibrium entropy , reached when @xmath32 . instead of @xmath33 , where @xmath34 is the time 
dependent parameter in the hamiltonian . ]
due to relaxation @xmath35 becomes smaller and smaller and at the end ( for a fixed potential ) @xmath36 .
to proceed we have to compute the distribution function . because the analytical integration of the kinetic equation for time  dependent potentials ( even with the simple relaxation time ansatz ) in general is not possible
we look for a perturbation theory approach .
there are two interesting limiting cases as starting points for this attempt : an infinitely slow potential change , corresponding to the adiabatic time evolution of the distribution function , and the relaxation less time evolution [ @xmath37 in eq .
] , corresponding to the liouville evolution ( denoted as @xmath38 ) , respectively .
we choose here the second way , where the system has a large relaxation time , compared with the time @xmath2 , for which we are interested , so that @xmath39 . an analytical integration of the time 
dependent collision less be is still impossible in general , but here we do not need the solution explicitly .      the lowest ( zeroth )
order of such an expansion is the time 
dependent liouville function @xmath38 .
we split the true distribution function into two parts , @xmath40 where @xmath41 is the solution of the collision less be and @xmath42 is the deviation from the actual distribution .
if we enter with this ansatz into eq .  , we get for the first order correction @xmath43~.\ ] ] this first order expression is sufficient as long as the ratio @xmath44 is small .
the equilibrium distribution @xmath45 is computed using @xmath38 on the right hand side of the conditions eq .  , eq .  .
we do not require the computation of @xmath46 , but we know that it does not depend on the relaxation time @xmath47 .
the distribution function eq .   with eq .
allows us to clarify the influence of @xmath47 on the dissipated energy .
we consider the begin of a potential change , where the distribution function deviates weakly from equilibrium .
the starting point is eq .  , where we replace @xmath27 , using eq .
, by @xmath48 now we use @xmath3 from eq .  , where we approximate the time integral in eq .
by a trapezoidal rule .
after some simplifications we obtain up to the order @xmath49 , @xmath50\int \frac{(f_l - f_g^{(0)})^2}{2 f_g^{(0)}}\mathrm{d}\gamma~.\ ] ] the dissipated energy ( heat ) is given by @xmath51 $ ] . for small times
the choice @xmath52 is reasonable and we get means the temperature change for an adiabatic process .
it is also possible to set here the initial temperature @xmath53 or a weighted sum of both . ]
@xmath54\int \frac{(f_l - f_g^{(0)})^2}{2 f_g^{(0)}}\mathrm{d}\gamma~.\ ] ] for @xmath55 we obtain from this expression @xmath56 and further ( for @xmath39 ) @xmath57 this result shows that the time 
dependent energy @xmath58 represents an upper bound of the energy of the gas , independent of the characteristics of the potential .
relaxation favors minimization of dissipation .
this is remarkable because the liouville evolution itself conserves entropy @xmath59 . for larger times
eq .   can be violated , however for slow potential changes it must fulfilled , because the true distribution functions @xmath3 and @xmath20 approaches the adiabatic distribution function @xmath60 . because the dissipated energy is given by @xmath61 one obtains @xmath62 .
for a model potential we consider a one dimensional harmonic oscillator with time 
dependent frequency , as widely used to investigate dissipation of brownian particles and the jarzynski equation @xcite .
an increase / decrease of the frequency leads to a compression / expansion of the gas .
we integrate numerically the be with the relaxation time ansatz and monitor the energy ( and entropy ) of the gas during the potential change .
the numerical method is based on a discretization of the phase space into equal sized cells and a numerical integration of the resulting ordinary differential equations . to obtain the time 
dependent equilibrium distribution @xmath20 , we compute for each time step the integrals eq .   and eq .   and solve numerically the resulting equations for @xmath63 and @xmath64 ( where the values for the quasistatic potential change are used as a starting guess ) .
this procedure runs well and there are no numerical problems which we should report . to be sure that energy conservation ( not only number of particles ) for a fixed potential is fulfilled , the used numerical limits of the phase space
must be large enough .
the external potential has the form @xmath65 here @xmath66 characterize the switching time and likewise denotes the final integration time .
we set @xmath67 as our time scale .
we choose @xmath68 and @xmath69 , so we are far away from adiabatic conditions .
the mean free fly time of a particle is on the order @xmath70 , where we have choosen @xmath71 ( and the ratio initial temperature to initial frequency @xmath72 ) . for bose gases at low temperatures and densities as h and na in magneto  optical traps
the relaxation time differs dramatically due to the different @xmath73-wave scattering cross sections ( by a factor of @xmath74)@xcite . for an adiabatic potential change the temperature of the gas
is given by @xmath75 and the energy @xmath76 , respectively the energy changes as @xmath77 $ ] . ] .
( color online ) ( a ) energy of the gas during the compression and expansion .
the frequency ratio is @xmath78 .
the green line corresponds to the liouville evolution ( @xmath58 ) and the blue line to a evolution with finite relaxation time ( @xmath79 ) .
the red dotted line is the adiabatic limit and in black  dotted it is shown the result from the perturbation theory .
( b ) shows the dissipated energy for a time regime where eq .
is valid . in ( c )
we have plotted the entropy of the gas and the approximation according eq .  .
for small times @xmath58 is the upper bound for the energy . , height=60 ]    ( color online ) energy of the gas during the expansion and compression cycle , @xmath80 .
although it is difficult to see the energy corresponding to the liouville evolution @xmath58 for small times is always greater than all others , in accordance to eq .
.,height=60 ]    ( color online ) the same potential deformation as in fig .
[ fig1 ] but 3.5 times slower as there ( with @xmath81 ) . in ( a )
it is shown the initial maxwell  boltzmann distribution in phase space [ in units @xmath82 , ( b ) shows the partially relaxed distribution at @xmath83 and ( c ) shows the liouville distribution @xmath84 . in ( d )
we have plotted again the energy of the gas ( blue line ) to demonstrate the validity of eq .   for the entire time.,height=70 ]    in fig .
[ fig1 ] we show the energy / entropy for one compression 
expansion chicle as a function of time for a gas with specified relaxation time .
the lines for the energy reflect the time 
dependent frequency , whereas the entropy increase as required from the h  theorem .
we have also plotted the perturbation theory expressions eq .   and eq .   to demonstrate the result eq
[ fig2 ] demonstrates the same behavior of the gas for the opposite chicle , where first the frequency @xmath85 decrease and hereafter increase ( expansion and compression ) .
here we have plotted the time 
dependent energy of two gases with different relaxation times and additional @xmath86 and @xmath87 . for small times
@xmath87 is an upper bound of the energy .
[ fig3 ] shows the distribution function , obtained by the numerical integration .
the initial maxwell  boltzmanm distribution ( a ) and the collision less evolved function ( c ) , as well as an example with finite relaxation time ( b ) the are plotted . in this case
( slower potential change as in the previous example ) the energy of the @xmath88 system is smaller as @xmath58 for the entire switching time .
in this paper we have discussed energy dissipation in gases in time 
dependent trapping potentials . using the boltzmann equation with the relaxation time ansatz we have shown that the dissipated energy depends on the relaxation time . in particular
we have demonstrated using perturbation theory , that for small times ( compared with the relaxation time ) the relaxation less time evolution always leads to a higher energy than that with finite relaxation time . on this time
scales relaxation decreases dissipation .
the derivation shows that this statement is fairly general : it does not depend on the shape of the external potential
. it would be interesting to make some numerical studies with the full collision integral of the be to check if the results of this paper are still valid @xcite .
the time derivative of the entropy in the relaxation time ansatz is equal @xmath89 if we set inside the logarithm @xmath90 we obtain @xmath91 \mathrm{d}\gamma~.\end{aligned}\ ] ] now we split the argument of the logarithm in to two parts , @xmath92 \mathrm{d}\gamma~,\ ] ] where we have defined the equilibrium entropy @xmath93 .
because @xmath20 is the equilibrium distribution , corresponding to the same number of particles and energy as @xmath3 , it follows @xmath94 .
the last integral in eq .
is positive for all @xmath2 .
if @xmath95 is a real number , then @xmath96 is always true . using this for the integral above we obtain ( @xmath97 ) , @xmath98 \mathrm{d}\gamma\leq \int \delta f_g \mathrm{d}\gamma=0~.\ ] ] with the @xmath99 sign in front of the integral in eq .
we conclude that @xmath100 .
+ in the text we use an expression for @xmath101 for small @xmath102 .
one can obtain this expression by insertion of @xmath28 into the logarithm in eq .   and a series expansion , @xmath103
we investigate an ideal gas in a time  
 dependent external trapping potential . 
 we use the boltzmann equation with the relaxation time ansatz to explore the time  
 dependent energy of an adiabatically isolated system . 
 in particular we are interested on the dissipation during a potential change along a given protocol with finite velocity . 
 the role of the relaxation less evolution as a limiting case is studied : starting from an equilibrium distribution and for small times the energy of the gas with relaxation is always smaller than that of the relaxation  less . 
 this means that relaxation processes show an ambivalent behavior : on the one hand entropy production , but on the other hand reduction of dissipation by driving back the system into equilibrium .