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In this paper, we first obtain the ($n+1$)-dimensional dilaton-Lifshitz blackhole (BH) solutions in the presence of Born-Infeld (BI) electrodynamics. Wefind that there are two different solutions for $z=n+1$ and $z\neq n+1$ cases($z$ is dynamical critical exponent). We show that the thermodynamics first lawis satisfied for both cases. Then, we turn to study different phase transitions(PTs) for our BHs. We start with study of Hawking-Page PT for both linearly andBI charged BHs. After that, we discuss the PTs inside the BHs. We present theimproved Davies quantities and prove that the PT points shown by them coincidewith Ruppeiner ones. We show that the zero temperature PTs are transitions onradiance properties of BHs by using Landau-Lifshitz theory. Next, we turn tostudy Ruppeiner geometry of linearly and BI charged BHs. For linearly chargedcase, we show that there are no PT at finite temperature for the case $z\geq2$. For $z<2$, it is found that the number of finite temperature PT pointsdepends on the value of BH charge and is not more than two. When we have twofinite temperature PT points, there are no thermally stable BH between thesetwo points and we have discontinues small/large BH PTs. As expected, for smallBHs, we observe finite magnitude for Ruppeiner invariant which shows the finitecorrelation between possible BH molecules while for large BHs, the correlationis very small. Finally, we study the Ruppeiner geometry and thermal stabilityof BI charged Lifshtiz BHs for different values of $z$. We observe that smallBHs are thermally unstable in some situations. Also, the behavior ofcorrelation between possible BH molecules for large BHs is the same as linearlycharged case. In both linearly and BI charged cases, for some choices ofparameters, the BH systems behave like a Van der Waals gas near transitionpoint.

european physical journal. c, particles and fields

Thermodynamics, phase transitions and Ruppeiner geometry for  Einstein-dilaton Lifshitz black holes in the presence of Maxwell and  Born-Infeld electrodynamics