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Some objects, especially metallic ones, feel cold on touching and others like wood, etc. feel warm on touching. Both are exposed to same environment and are in their stable state, so some kind of equilibrium must be being reached. What is this equilibrium?
And how do we perceive hotness or coldness of an object? Does skin have some kind of heat sensors, etc., which transmit signals to brain? Like, how do the eye transmit/convey an image formed on the retina to brain?
Think of the layer of skin on the finger tip which is in contact with the object that has a lower temperature than the body. If the object has a high heat capacity and a high thermal conductivity (like metal), then the skin of the finger will come to an equilibrium temperature that is lower than if it is in contact with an object that has a lower heat capacity and a lower conductivity (like wood). So the nerves in the finger tip are detecting temperature, but it is the temperature of the layer of skin that is being measured, not the temperature of the object.
In other words, the body is trying to keep the skin warm with blood flow and by conduction of heat, but the metal is better able to lower the temperature of the skin than wood can.
Throughout the outer body shell (ectoderm), encompassing skin and subcutaneous fat tissue, thermoceptors are embedded on free nerve fibres (so, these types of receptors are not special cell types, in contrast to photoreceptors for example). There are two types of fibres: group-III afferences (cold-sensitive) and group-IV afferences (heat-sensitive). In the membranes of these nerve cell fibres are specific proteins anchored. Those proteins are called TRP channels (Transient receptor potential), which can conduct ions on demand.
When temperature difference on the skin periphery is detected, they change their conformation and open themselves which leads to the influx of Na+, Cal2+ and Mg2+ ions leading to action potentials transmitted to the central nervous system. Those types of thermoceptors work like proportional-differential-sensors: rapid skin temperature changes (touching a warm cup of coffee) are answered with very high action potential frequency; afterwards they adapt to the temperature and fire less frequent, but still more frequent than in default. This kind of firing leads to the impression of holding something in the hand which is warmer than one's own skin before touching it.
Now, very interestingly, TRP receptors can be activated via capsaicin (chili spicy molecule), allicin (garlic aroma molecule), allyl isothiocyanat (wasabi compound), menthol or even THC. This is the reason why chili can evoke a real sensation of hotness while menthol evokes the impression of coldness when you for example taste a menthol chewing gum.
With those biological devices one person can perceive temperature differences of 0.2 K at best.
The elaborate mechanism for heat detection in human skin is mainly carried by somatosensory system, which is responsible for temperature, touch, pressure sensing.
Somatosensory system contains thermoreceptors which are chiefly responsible for thermo sensing. wiki
These thermoreceptors are able to detect absolute, to a certain extent and relative, mainly temperatures.
Their functioning is attibuted by complex ion conduction paths of $Na^+$ and $Ca^{2+}$.
I don't know the actual biological process, but I know that our body detects heat transfer, not temperature. That's why an object with a higher heat transfer coefficient (such as metallic objects) appear colder than wood at the same temperature, for example.
To the first part of your question: What is this equilibrium?
At room temperature the metallic or wooden object is in equilibrium with it's environment. When you touch the object this equilibrium is disturbed. As the skin on your fingers will be quite a bit above room temperature when you touch the object heat will flow from your finger to the object. The amount of heat depends on the contact area, force with which you touch the object etc. Your finger will get colder, as the heat is flowing into the object.
An equilibrium is reached again, when this heat flow does not change over time. This can mean that the object has the same temperature as your skin or that the heat is going into the environment at a constant rate through the object.
The cold or warm feel is depending on the amount of heat that flows out of your finger into the object and is sensed by the actual temperature change of your finger. The biological mechanism has been described by Vineet Menon nicely.
sensation of heat activates an electrical impulse which is passed through neurons and synapsis as electrical an chemical reactions and reaches our brain which interprets the message as heat some times if the heat is unbearable the message is interpreted by the spinal chord
