  A novel collisionless shock jump condition is suggested by modeling the
entropy production at the shock transition region. We also calculate downstream
developments of the atomic ionization balance and the ion temperature
relaxation in supernova remnants (SNRs). The injection process and subsequent
acceleration of cosmic-rays (CRs) in the SNR shocks are closely related to the
formation process of the collisionless shocks. The formation of the shock is
caused by wave-particle interactions. Since the wave-particle interactions
result in energy exchanges between electromagnetic fields and charged
particles, the randomization of particles associated with the shock transition
may occur with the rate given by the scalar product of the electric field and
current. We find that order-of-magnitude estimates of the randomization with
reasonable strength of the electromagnetic fields in the SNR constrain the
amount of the CR nuclei and ion temperatures. The constrained amount of the CR
nuclei can be sufficient to explain the Galactic CRs. The ion temperature
becomes significantly lower than in the case of no CRs. To distinguish the case
without CRs, we perform synthetic observations of atomic line emissions from
the downstream region of the SNR RCW~86. Future observations by {\it XRISM} and
{\it Athena} can distinguish whether the SNR shock accelerates the CRs or not
from the ion temperatures.
