Thermodynamics, temperature below 0 Kelvin I read a news article about how they were able to create a negative temperature, below absolute zero, and my question is how does this work?
I know that there are different definitions of temperature, and for the negative temperature, it's referring to entropy, the disorder.
Can someone explain to me how is the negative temperature created and why is it in some situations when we add energy to a system, entropy does not increase?
 A: Spontaneous process
Spontaneous process is a process which can take place by itself under the given set of conditions once it has been initiated if necessary.
For example, A cup of hot tea kept in your room cools spontaneously by losing heat to the surroundings. Another classic example is galvanic cell, which converts chemical energy produced during spontaneous redox reactions into electrical energy.
Non-Spontaneous process
Non-Spontaneous process is the process which is made to occur only by supplying energy continuously from outside the system.
For example, a cup of tea can be made hot (reverse of natural process) by heating. Electrolytic cell is also an example in electrical energy is used to bring about non-spontaneous reactions.
Gibbs free energy
If gibbs free energy is $-VE$, process is said to be spontaneous or else it is non-spontaneous. Gibbs free energy is given by
$$\triangle{G}=\triangle{H}-T\triangle{S}$$
where, $G$ is gibb's energy, $H$ is enthalpy of the system, $S$ is entropy (randomness or disorderness) of the system.
Following sign conventions are used in the below explanation,

*

*Heat is absorbed: $\triangle{H}=+VE$.

*Disordeness is increased: $\triangle{S}=+VE$
$-VE$ signs are used for other cases not mentioned.

Can someone explain to me how is the negative temperature created and why is it in some situations when we add energy to a system, entropy does not increase?

Generally external energy is supplied to create negative temperature. So, gibbs energy is $+VE$.
$\triangle{G}(+VE)=\triangle{H}(+VE)-T(-VE)\triangle{S}(+VE)$
From the above equation it follows that, if a system is made to absorb heat and if its disorderness is increased by the non-spontaneous reaction, temperature can be made negative. Or else entropy can be made $-VE$, but it requires excess of heat to be absorbed by the system.
In some cases energy you have supplied is completely absorbed by the system (assuming temperature to be not zero kelvin), i.e $\triangle{G}=\triangle{H}$ then entropy will not increase i.e $\triangle{S}=0$
