5.3 Matrix Distribution from a QFT Density Matrix
95
However, as we saw in Sect. 5.2, we will only consider the cases where m s = m s in
any non-trivial block of f ss , so in what follows N
p,ss will be constant in T .
Following the standard argumentation for deriving the Boltzmann equation, the
statistical state in the “in” region can be assumed to be minimally correlated, i.e. it
is entirely determined by the corresponding 1-particle distribution f
in
ss (
p) instead of
a full BBGKY-like hierarchy. More precisely, the higher order moments
∼ ∼a
†
p 1 ,s 1
. . . a
†
p n ,s n
a
q 1 ,r 1 . . . a
q m ,r m ρ in ,
(5.3.24)
factorize into products of the two-point functions
a
†
p ,s a
p,s ρ in ≡ (2π)
3
δ
(3)
(
p − −
p
) f
in
ss (
p) ,
(5.3.25)
a
p,s a
†
p ,s ρ in ≡ (2π)
3
δ
(3)
(
p − −
p
)
δ ss + (−1)
|s||s
| f
in
ss (
p)
, (5.3.26)
where the first equation is the definition of f
in
ss (
p), while the second one is obtained by
using Eq. (5.3.8). This is the assumption of “molecular chaos” in the quantum context,
by which the particle momenta are uncorrelated before scattering and thus f
in
ss (
p) is
a complete enough description of the state. This will not hold in general for the “out”
state ρ out , i.e. it will not be expressible solely in terms of its 1-particle distribution
f
out
ss (
p), because the collision will correlate the outcoming states. Nevertheless, since
the gas is dilute, the macroscopic free evolution between two successive scattering
events is long enough to make the higher-order correlation functions decay, thus
leading again to an uncorrelated “in” statistical state for the next scattering event. This
unequal treatment of the “in” and “out” regions breaks the time-reversal symmetry
and thus generates the “arrow of time” at the mesoscopic and macroscopic levels.
Thus, the distribution f ss (x,
p) that will ultimately obey the Boltzmann equation is
f
in
ss (x,
p). For this reason, from now on we focus on the “in” region and simplify
the notation to ρ := ρ in and f ss (
p) := f
in
ss (
p).
5.4 Expressing ρ and the Entropy Current in Terms of f ss
Since ρ(x) is entirely determined by f ss (x,
p), the relation in Eq. (5.3.20) can be
inverted, subject to the conditions (5.3.2) and (5.3.17). In particular, this will allow
us to express the entropy density
s(x) := −V
−1 Tr
ρ(x) log ρ(x)
,
(5.4.1)
in terms of f ss (x,
p). Given that the only non-trivial operator involved in Eq. (5.3.20)
is N
p,ss , the inversion must take the form
ρ(x) = Z
−1
(x) F
−
d
3 p
(2π) 3 w ss (x,
p) N
p,ss
,
(5.4.2)
95
However, as we saw in Sect. 5.2, we will only consider the cases where m s = m s in
any non-trivial block of f ss , so in what follows N
p,ss will be constant in T .
Following the standard argumentation for deriving the Boltzmann equation, the
statistical state in the “in” region can be assumed to be minimally correlated, i.e. it
is entirely determined by the corresponding 1-particle distribution f
in
ss (
p) instead of
a full BBGKY-like hierarchy. More precisely, the higher order moments
∼ ∼a
†
p 1 ,s 1
. . . a
†
p n ,s n
a
q 1 ,r 1 . . . a
q m ,r m ρ in ,
(5.3.24)
factorize into products of the two-point functions
a
†
p ,s a
p,s ρ in ≡ (2π)
3
δ
(3)
(
p − −
p
) f
in
ss (
p) ,
(5.3.25)
a
p,s a
†
p ,s ρ in ≡ (2π)
3
δ
(3)
(
p − −
p
)
δ ss + (−1)
|s||s
| f
in
ss (
p)
, (5.3.26)
where the first equation is the definition of f
in
ss (
p), while the second one is obtained by
using Eq. (5.3.8). This is the assumption of “molecular chaos” in the quantum context,
by which the particle momenta are uncorrelated before scattering and thus f
in
ss (
p) is
a complete enough description of the state. This will not hold in general for the “out”
state ρ out , i.e. it will not be expressible solely in terms of its 1-particle distribution
f
out
ss (
p), because the collision will correlate the outcoming states. Nevertheless, since
the gas is dilute, the macroscopic free evolution between two successive scattering
events is long enough to make the higher-order correlation functions decay, thus
leading again to an uncorrelated “in” statistical state for the next scattering event. This
unequal treatment of the “in” and “out” regions breaks the time-reversal symmetry
and thus generates the “arrow of time” at the mesoscopic and macroscopic levels.
Thus, the distribution f ss (x,
p) that will ultimately obey the Boltzmann equation is
f
in
ss (x,
p). For this reason, from now on we focus on the “in” region and simplify
the notation to ρ := ρ in and f ss (
p) := f
in
ss (
p).
5.4 Expressing ρ and the Entropy Current in Terms of f ss
Since ρ(x) is entirely determined by f ss (x,
p), the relation in Eq. (5.3.20) can be
inverted, subject to the conditions (5.3.2) and (5.3.17). In particular, this will allow
us to express the entropy density
s(x) := −V
−1 Tr
ρ(x) log ρ(x)
,
(5.4.1)
in terms of f ss (x,
p). Given that the only non-trivial operator involved in Eq. (5.3.20)
is N
p,ss , the inversion must take the form
ρ(x) = Z
−1
(x) F
−
d
3 p
(2π) 3 w ss (x,
p) N
p,ss
,
(5.4.2)