New Jersey 101.5's, "What to know about a second marriage and an inheritance," discusses what children can expect when a father passes away when married to his second wife.
When someone dies, they typically will have two types of property at death. These are probate assets and non-probate assets.
Probate assets are those that a person owns alone without a named beneficiary. The probate assets will pass in accordance with the terms of a person's will. However, if there's no will, the assets pass according to the state's laws of intestacy. Non-probate assets include assets that a person owns jointly with another person, like a home or a joint checking accounts, and assets which designate a beneficiary. These are things like life insurance and retirement assets. Non-probate assets pass to the joint survivor or the named beneficiary by operation of law. These aren't governed by the person's will or the laws of intestacy (unless the named beneficiary is the estate).
In one example, if the father died with probate assets, his estate would be required to be administered through the probate court in the county of his residence at the time of his death. If the father had a will, it's presented for probate. The person named in the will as the executor can qualify and obtain Letters Testamentary—these authorize him or her to act on behalf of the estate. If there isn't a will, an administrator is appointed. Generally, the spouse and then the children will have the first right to such an appointment. In either situation, everyone named in the will and all heirs at law (which include children) must be given notice of probate. They can also requested and get a copy of the will.
A person can choose to leave nothing to a spouse, but a spouse is typically entitled to claim an elective share. In many states, the spouse receives one third of a decedent's estate.
If the father died without a will and if there was no premarital agreement, then assets pass in intestacy where the surviving spouse will receive the first 25% under some state laws. The rest will pass to the children (and the children of deceased siblings), if any.
Reference: New Jersey 101.5 (March 29, 2016) "What to know about a second marriage and an inheritance"