  Till now, critical behaviour of Lifshitz black holes, in an extended $P-v$
space, has not been studied, because it is impossible to find an analytical
equation of state, $P=P(v,T)$, for an arbitrary Lifshitz exponent $z$. In this
paper, we adopt a new approach toward thermodynamic phase space and
successfully explore the critical behaviour of $(n+1)$-dimensional Lifshitz
dilaton black holes. For this purpose, we write down the equation of state as
$Q^s=Q^s(T,\Psi)$ with $\Psi=\left({\partial M}/{\partial Q^{s} }\right)_{S,P}$
is the conjugate of $Q^s$ and construct Smarr relation based on this new phase
space as $ M=M(S,Q^{s},P)$, where $s=2p/(2p-1)$ with $p$ is the power of the
power-law Maxwell Lagrangian. We justify such a choice mathematically and show
that with this new phase space, the system admits the critical behaviour and
resembles the Van der Waals fluid system when the cosmological constant
(pressure) is treated as a fixed parameter, while the charge of the system
varies. We obtain Gibbs free energy of the system and find swallow tail shape
in Gibbs diagrams which represents the first order phase transition. Finally,
we calculate the critical exponents and show that although thermodynamic
quantities depend on the metric parameters such as $z$ , $p$ and $n$, the
critical exponents are the same as Van der Walls fluid-gas system. This
alternative viewpoint toward phase space of lifshitz dilaton black hole can be
understood easily since one can imagine such a change for a given single black
hole i.e., acquiring charge which induces the phase transition. Our results
further support the viewpoint suggested in \cite{Dehy}.
