Tenant Rights After Notice to Quit & Double Rent

Am I right in thinking that if one tenant leaves after giving more than four weeks' written notice and at the end of a rental month, the remaining tenant refusing to leave becomes a trespasser?  As a trespasser, damages are awarded equivalent to double the rent. 

Also, as the remaining tenant is a trespasser, I can contact the police on the day to see if the police will remove him from the property.  In addition, on the property is another person who is not a permitted occupier. 

I have copied and pasted notes I found online that I believe are relevant.

Section 18 Distress for Rent Act 1737

Section 18 entitles the landlord to double the passing rent where the tenant has given a notice that he will quit the premises at a specified time but subsequently continues to occupy the premises in contravention of the notice. 

The failure of the tenant to vacate on the expiry of its notice to quit triggers the landlord’s entitlement to double rent under the 1737 Act. The double rent becomes chargeable for the period of “holding over” by the tenant, i.e. from the expiry of the notice until the day the tenant actually vacates.

For section 18 to apply, the landlord must treat the former tenant as a trespasser in unlawful occupation and must not do or say anything which would treat the lease as continuing. 

It is therefore essential that the landlord does not accept rent at the old level once the notice to quit has expired otherwise he will be held to have waived his right to claim double rent.

The object of the 1737 Act is to compensate the landlord for potential loss of rent when the tenant held over against the landlord’s insistence that the tenant should comply with his notice to quit. Therefore, similarly to the 1730 Act, the right to double rent arises only when the tenant holds over after he has served a notice to quit and, therefore, is in fact a trespasser and the landlord treats him as such.

Since 2012 under s144 of the Legal Aid Sentencing and Punishment of Offenders Act 2012 landlords can now notify the police that their property is occupied by trespassers as this is now a criminal offence so do not necessarily need to use the civil court system. However, much will depend on the willingness of the police to intervene and, where the police are not willing to make arrests, landlords will need to follow the civil law rules which are still in force.